the quantum theory of magnetism

TheQuantum Theoryof

MagnetismS e c o n d E d i t i o n

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N o r b e r t o M a j l i s McGill University, Canada

N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

World Scientific

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THE QUANTUM THEORY OF MAGNETISM (2nd Edition)

Alvin - The Quantum Theory.pmd 11/23/2007, 9:00 AM1

Preface to the SecondEdition

For the second edition I have added three new chapters, namely on magneticanisotropy, on coherent magnon states and on local moments.

Besides, I have enlarged the chapter on itinerant magnetism, by including asection on paramagnons.

I have corrected several errors which appeared in the first edition. Some ofthese were indicated to me by friends and colleagues, for which I thank all ofthem.

Norberto Majlis

e-mail: [email protected]

Centre for the Physics of Materials, McGill University

Montreal, April 2007.

v

Preface to the First Edition

This is an advanced level textbook which grew out of lecture notes for sev-eral graduate courses I taught in different places over several years. It assumesthat the reader has a background of Quantum Mechanics, Statistical Mechanicsand Condensed Matter Physics. The methods of Green’s functions, which arestandard by now, are used fairly extensively in the book, and a mathematicalintroduction is included for those not very familiar with them.

The selection of subjects aims to present a description of the behaviour ofsystems which show ordered magnetic phases. This, plus the necessary limita-tion of the extension within reasonable limits, imposed the exclusion of manyimportant subjects, among them diamagnetism, the Kondo effect, magnetic res-onance, disordered systems, etc.

In turn, the reader will find a detailed presentation of the mean-field approx-imation, which is the central paradigm for the phenomenological description ofphase transitions, a discussion of the properties of low-dimensional magneticsystems, a somewhat detailed presentation of the RKKY and related models ofindirect exchange and a chapter on surface magnetism, among other character-istics which make it different from other texts on the subject.

This book can be used as a text for a graduate course in physics, chemistry,chemical engineering, materials science and electrical engineering and as a ref-erence text for researchers in condensed matter physics.

Many exercises are included in the text, and the reader is encouraged to takean active part by trying to solve them.

I hope readers who find errors in the book or want to suggest improvementsget in touch with me.

It is a great pleasure to acknowledge the moral support I enjoyed from

vii

viii PREFACE TO THE FIRST EDITION

my wife and my children, the generous and extremely competent help with thesoftware during the preparation of the manuscript from Luis Alberto Giribaldoand my daughter Flavia, and the careful reading of several chapters by JuneGoncalves. I want also to express deep recognition to the Centre for the Physicsof Materials of McGill University for their support, particularly to Martin J.Zuckermann, Martin Grant and Juan Gallego.

Norberto Majlis

Centre for the Physics of Materials, McGill University

Montreal, March 7th, 2000.

Contents

1 Paramagnetism 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Quantum mechanics of atoms . . . . . . . . . . . . . . . . . . . . 4

1.2.1 L-S (Russel-Saunders) coupling . . . . . . . . . . . . . . . 41.2.2 Hund’s rules . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Spin-orbit splitting . . . . . . . . . . . . . . . . . . . . . . 7

1.3 The quantum theory of paramagnetism . . . . . . . . . . . . . . 81.4 Crystal-field corrections . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 Effects of crystal-field symmetry . . . . . . . . . . . . . . 121.4.2 Stevens operator equivalents . . . . . . . . . . . . . . . . 14

1.5 Quenching of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.6 Time reversal and spin . . . . . . . . . . . . . . . . . . . . . . . . 20

1.6.1 Kramers degeneracy . . . . . . . . . . . . . . . . . . . . . 211.7 Effective spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 22

1.7.1 Effective gyromagnetic ratio . . . . . . . . . . . . . . . . . 231.7.2 Single-ion anisotropy energy . . . . . . . . . . . . . . . . . 23

2 Interacting Spins 272.1 Weiss model of ferromagnetism . . . . . . . . . . . . . . . . . . . 27

2.1.1 Critical behaviour of Weiss model . . . . . . . . . . . . . 282.2 Microscopic basis of magnetism . . . . . . . . . . . . . . . . . . . 29

2.2.1 The direct exchange interaction . . . . . . . . . . . . . . . 302.2.2 The superexchange mechanism . . . . . . . . . . . . . . . 362.2.3 The RKKY interaction . . . . . . . . . . . . . . . . . . . 40

3 Mean Field Approximation 433.1 Helmholtz free energy . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Mean field susceptibility . . . . . . . . . . . . . . . . . . . . . . . 503.3 Specific heat of ferromagnet . . . . . . . . . . . . . . . . . . . . . 523.4 The Oguchi method . . . . . . . . . . . . . . . . . . . . . . . . . 553.5 Modulated phases . . . . . . . . . . . . . . . . . . . . . . . . . . 583.6 MFA for antiferromagnetism . . . . . . . . . . . . . . . . . . . . . 61

3.6.1 Longitudinal susceptibility . . . . . . . . . . . . . . . . . . 633.6.2 Transverse susceptibility . . . . . . . . . . . . . . . . . . . 65

ix

x CONTENTS

3.6.3 Spin-flop and other transitions . . . . . . . . . . . . . . . 683.7 Helimagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.8 Goldstone’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Spin Waves 794.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Holstein-Primakoff transformation . . . . . . . . . . . . . . . . . 824.3 Linear spin-wave theory . . . . . . . . . . . . . . . . . . . . . . . 844.4 Semiclassical picture . . . . . . . . . . . . . . . . . . . . . . . . . 854.5 Macroscopic magnon theory . . . . . . . . . . . . . . . . . . . . . 884.6 Thermal properties . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.6.1 Total spin deviation . . . . . . . . . . . . . . . . . . . . . 894.6.2 Non-linear corrections . . . . . . . . . . . . . . . . . . . . 91

4.7 The Heisenberg antiferromagnet . . . . . . . . . . . . . . . . . . 924.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 924.7.2 Antiferromagnetic spin-waves . . . . . . . . . . . . . . . . 934.7.3 Sublattice magnetization . . . . . . . . . . . . . . . . . . 974.7.4 Ground state energy of AFM . . . . . . . . . . . . . . . . 99

5 Magnetic Anisotropy 1015.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2 Microscopic origin of anisotropy . . . . . . . . . . . . . . . . . . . 1045.3 Magneto-elastic coupling . . . . . . . . . . . . . . . . . . . . . . . 1085.4 Magnetostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.5 Inverse magnetostriction . . . . . . . . . . . . . . . . . . . . . . . 1135.6 Induced magneto-crystalline anisotropy . . . . . . . . . . . . . . 115

6 Green’s Functions Methods 1176.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2 Spectral representation . . . . . . . . . . . . . . . . . . . . . . . . 1196.3 RPA for spin 1/2 ferromagnet . . . . . . . . . . . . . . . . . . . . 1226.4 Comparison of RPA and MFA . . . . . . . . . . . . . . . . . . . . 1266.5 RPA for arbitrary spin . . . . . . . . . . . . . . . . . . . . . . . . 1276.6 RPA for ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . 129

6.6.1 Paramagnetic phase . . . . . . . . . . . . . . . . . . . . . 1296.6.2 Linear FM chain . . . . . . . . . . . . . . . . . . . . . . . 1326.6.3 Square FM lattice . . . . . . . . . . . . . . . . . . . . . . 133

6.7 FM with a finite applied field . . . . . . . . . . . . . . . . . . . . 1356.8 RPA for antiferromagnet . . . . . . . . . . . . . . . . . . . . . . . 136

6.8.1 Spin 1/2 AFM . . . . . . . . . . . . . . . . . . . . . . . . 1376.8.2 Arbitrary spin AFM . . . . . . . . . . . . . . . . . . . . . 1386.8.3 Zero-point spin deviation . . . . . . . . . . . . . . . . . . 1396.8.4 Correlation length . . . . . . . . . . . . . . . . . . . . . . 140

6.9 RPA susceptibility of AFM . . . . . . . . . . . . . . . . . . . . . 1436.10 Spin-flop transition . . . . . . . . . . . . . . . . . . . . . . . . . 1466.11 χ‖ at low T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

CONTENTS xi

6.12 Transverse susceptibility . . . . . . . . . . . . . . . . . . . . . . . 1496.13 Single-site anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . 1506.14 Dynamic linear response . . . . . . . . . . . . . . . . . . . . . . . 1536.15 Energy absorbed from external field . . . . . . . . . . . . . . . . 1566.16 Susceptibility of FM . . . . . . . . . . . . . . . . . . . . . . . . . 1586.17 Corrections to RPA . . . . . . . . . . . . . . . . . . . . . . . . . 158

7 Dipole-Dipole Interactions 1617.1 Dipolar Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 1617.2 Dipole-exchange spin-waves . . . . . . . . . . . . . . . . . . . . . 1667.3 Uniform precession (k = 0) mode . . . . . . . . . . . . . . . . . . 1707.4 Eigenmodes for k 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . 174

7.4.1 Demagnetization factors . . . . . . . . . . . . . . . . . . . 1767.5 Ellipticity of spin precession . . . . . . . . . . . . . . . . . . . . . 1767.6 Effect of magnons on total spin . . . . . . . . . . . . . . . . . . . 1777.7 Magnetostatic modes . . . . . . . . . . . . . . . . . . . . . . . . . 179

8 Coherent States of Magnons 1838.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.2 Coherent states of bosons . . . . . . . . . . . . . . . . . . . . . . 183

8.2.1 Overcompleteness of coherent states basis . . . . . . . . . 1858.3 Magnon number distribution function . . . . . . . . . . . . . . . 1868.4 Uncertainty relations . . . . . . . . . . . . . . . . . . . . . . . . . 1878.5 Phase states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1888.6 Magnon states of well defined phase . . . . . . . . . . . . . . . . 1908.7 Properties of the single-mode number states . . . . . . . . . . . . 1918.8 Properties of a single-mode phase state . . . . . . . . . . . . . . . 1928.9 Expectation value of local spin operators in a coherent state . . . 192

9 Itinerant Magnetism 1959.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.2 Pauli paramagnetic susceptibility . . . . . . . . . . . . . . . . . . 1959.3 Stoner model of ferromagnetic metals . . . . . . . . . . . . . . . 1969.4 Hubbard Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 1989.5 Instability of paramagnetic phase . . . . . . . . . . . . . . . . . . 1999.6 Magnons in the Stoner model . . . . . . . . . . . . . . . . . . . . 200

9.6.1 The RPA susceptibility . . . . . . . . . . . . . . . . . . . 2009.6.2 Singularities of the susceptibility . . . . . . . . . . . . . . 204

9.7 Tc in Stoner model . . . . . . . . . . . . . . . . . . . . . . . . . . 2089.8 Metals with degenerate bands . . . . . . . . . . . . . . . . . . . . 2109.9 Spin-density wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139.10 Hartree–Fock description of SDW . . . . . . . . . . . . . . . . . . 2159.11 Effects of correlations . . . . . . . . . . . . . . . . . . . . . . . . 222

9.11.1 Kinetic exchange interaction . . . . . . . . . . . . . . . . 2249.12 Paramagnetic instability and paramagnons . . . . . . . . . . . . 228

9.12.1 Paramagnon contribution to the specific heat . . . . . . . 232

xii CONTENTS

9.13 Beyond Stoner theory and RPA . . . . . . . . . . . . . . . . . . . 2349.14 Magnetism and superconductivity . . . . . . . . . . . . . . . . . 234

10 Indirect Exchange 23910.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23910.2 Effective s-d exchange interaction . . . . . . . . . . . . . . . . . . 24010.3 Indirect exchange Hamiltonian . . . . . . . . . . . . . . . . . . . 24710.4 Range function and band structure . . . . . . . . . . . . . . . . . 24810.5 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

10.5.1 Intrinsic semiconductors, high T . . . . . . . . . . . . . . 25610.5.2 Intrinsic semiconductors, low T . . . . . . . . . . . . . . . 25710.5.3 Degenerate semiconductors . . . . . . . . . . . . . . . . . 259

10.6 Magnetic multilayer systems . . . . . . . . . . . . . . . . . . . . . 259

11 Local Moments 26311.1 The s-d and Anderson Hamiltonians . . . . . . . . . . . . . . . . 26311.2 Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . 26311.3 Hartree–Fock solution of Anderson Hamiltonian . . . . . . . . . . 26411.4 Kondo effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

11.4.1 Calculation of resistivity . . . . . . . . . . . . . . . . . . . 26811.4.2 Calculation of the collision time . . . . . . . . . . . . . . . 26911.4.3 Higher order contributions . . . . . . . . . . . . . . . . . . 275

12 Low Dimensions 27712.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27712.2 Proof of Mermin–Wagner theorem . . . . . . . . . . . . . . . . . 278

12.2.1 Bogoliubov inequality . . . . . . . . . . . . . . . . . . . . 27812.2.2 Application to the Heisenberg model . . . . . . . . . . . . 279

12.3 Dipolar interactions in low dimensions . . . . . . . . . . . . . . . 28212.3.1 Dipole-exchange cross-over . . . . . . . . . . . . . . . . . 286

12.4 One dimensional instabilities . . . . . . . . . . . . . . . . . . . . 28712.5 Antiferromagnetic chain . . . . . . . . . . . . . . . . . . . . . . . 28912.6 RPA for the AFM chain . . . . . . . . . . . . . . . . . . . . . . . 291

12.6.1 Exchange dominated regime . . . . . . . . . . . . . . . . . 29112.6.2 Dipolar dominated regime . . . . . . . . . . . . . . . . . . 294

12.7 Dipolar interaction in layers . . . . . . . . . . . . . . . . . . . . . 29512.7.1 Monolayer . . . . . . . . . . . . . . . . . . . . . . . . . . . 29712.7.2 Bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

13 Surface Magnetism 30513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30513.2 MFA treatment of surfaces . . . . . . . . . . . . . . . . . . . . . . 30613.3 Surface excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 30913.4 LRPA method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31013.5 Wave functions for bulk and surface . . . . . . . . . . . . . . . . 31613.6 Surface density of magnon states . . . . . . . . . . . . . . . . . . 317

CONTENTS xiii

13.7 Surface phase-transitions . . . . . . . . . . . . . . . . . . . . . . . 31913.8 Dipolar surface effects . . . . . . . . . . . . . . . . . . . . . . . . 32313.9 Surface magnetism in metals . . . . . . . . . . . . . . . . . . . . 324

14 Two-Magnon Eigenstates 32914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32914.2 Green’s function formalism . . . . . . . . . . . . . . . . . . . . . 33114.3 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33714.4 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33914.5 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . 34014.6 Anisotropy effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

15 Other Interactions 34515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34515.2 Two-magnon interaction . . . . . . . . . . . . . . . . . . . . . . . 34715.3 Three-magnon processes . . . . . . . . . . . . . . . . . . . . . . . 34915.4 Magnon-phonon interaction . . . . . . . . . . . . . . . . . . . . . 35315.5 Bilinear magnon-phonon interaction . . . . . . . . . . . . . . . . 355

Appendix A Group Theory 359A.1 Definition of group . . . . . . . . . . . . . . . . . . . . . . . . . . 359A.2 Group representations . . . . . . . . . . . . . . . . . . . . . . . . 360

A.2.1 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . 361A.3 Orthogonality relations . . . . . . . . . . . . . . . . . . . . . . . . 362A.4 Projection operators . . . . . . . . . . . . . . . . . . . . . . . . . 362A.5 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . 363A.6 Wigner-Eckart theorem . . . . . . . . . . . . . . . . . . . . . . . 364A.7 Space groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365A.8 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

Appendix B Time Reversal 369B.1 Antilinear operators . . . . . . . . . . . . . . . . . . . . . . . . . 369B.2 Anti-unitary operators . . . . . . . . . . . . . . . . . . . . . . . . 371B.3 Time reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Index 375

Chapter 1

Paramagnetism

1.1 Introduction

Some examples of atomic systems with a permanent magnetic moment in theground state are:

• atoms, molecules, ions or free radicals with an odd number of electrons,like H,NO,C(C6H5)3, Na

+, etc.;

• a few molecules with an even number of electrons, like O2 and some organiccompounds;

• atoms or ions with an unfilled electronic shell. This case includes:

– transition elements (3d shell incomplete);

– the rare earths (series of the lanthanides) (4f shell incomplete);

– the series of the actinides (5f shell incomplete).

We shall consider in the rest of this chapter that the atomic entities carryingangular momenta occupy sites on a perfect crystalline insulator, that they arevery well localized on their respective sites, and that their mutual interactionsare negligible. This implies that we can neglect the unavoidable dipole-dipoleinteractions, which we assume are so weak that they could only affect the be-haviour of the system at extremely low temperatures. If such a system is placedin an external uniform magnetic field B the Zeeman energy term is:

V = −B ·N∑

i=1

µi (1.1)

where µi is the magnetic moment at site i. If the magnetic moments are replacedby classical vectors, as in the semi-classic, large J limit, the corresponding

1

2 CHAPTER 1. PARAMAGNETISM

partition function for this system in the canonical ensemble is:

Z =

∫

dΩ1dΩ2 · · · dΩN exp (βµ0B

N∑

i=1

cosθi) = (z(a))N (1.2)

where we have defined

z(a) =

∫

exp (a cosθ)dΩ =π

asinh(a) (1.3)

with a = βµ0B , µ0 = magnetic moment of each atom, β = 1/kBT , dΩi is thedifferential element of solid angle for the i-th dipole and θ is the angle betweenthat dipole and the applied magnetic field.

The Gibbs free energy per particle is:

f = −kBT log z (1.4)

and the average magnetic moment µ per atom along the applied field directionis, in units of µ0:

µ

µ0= − 1

µ0

∂f

∂B= coth(a) − 1/a ≡ L(a) (1.5)

here L(a) is the Langevin function [1].

Exercise 1.1Prove that for a 1, that is for B → 0 or T → ∞ or both, the magnetizationapproaches

mz ≈ (N/V )µ20B/3kBT (1.6)

(Curie’s law) [2].

Equation 1.5 describes Langevin paramagnetism. The typical value of µ0 isa few Bohr magnetons µB ≈ 10−20 erg gauss −1, so that only at very low tem-peratures or very high fields, like those produced with superconducting and/orpulsed refrigerated magnets, can saturation effects be observed in the mz(T )curve. We show in Fig. 1.1 a comparison of Langevin theory with experimentalmeasurements of the magnetization of Cr potassium alum. These measure-ments were performed in fields of up to 50, 000 gauss and at temperatures downto 1.29 K, which allowed for a large saturation degree. It is evident from Fig. 1.1that Langevin’s theory, which assumes continuity of the observable values of themagnetic dipolar moment of an ion or atom, does not fit the experiments, exceptat high temperatures and/or low fields.

In order to reach agreement with experiment one must incorporate thosechanges which are due to the quantum nature of the ions. The main effects ofangular momentum quantization are twofold:

1.1. INTRODUCTION 3

Figure 1.1: Plot of µ/µ0, in arbitrary units, vs. B/T for potassium chromiumalum. The heavy solid line is a Brillouin curve for g = 2 (complete quenchingof L) and J = S = 3/2 , fitted to experimental data at the highest value ofB/T . The thin solid line is a Brillouin curve for g = 2/5, J = 3/2 and L = 3(no quenching). The broken lines are Langevin curves fitted at the highest valueof B/T (lower curve) and at the origin (slope fitting) (upper curve). FromRef. [5].

• the discrete character of the eigenvalue spectrum of the vector compo-nents of angular momentum operators, or space quantization, leads to astatistical distibution for the magnetization different from that obtainedby Langevin. The consequence is the substitution of Langevin’s functionL(a) by Brillouin’s function BJ(a) (Sect. 1.3).

• the paramagnetic substances we are considering in this chapter are ioniccrystals which contain some ions with non-zero permanent magnetic mo-ment in the ground-state. In the solid, they have of course an electronicspectrum different from that of the free ion. The main effect of the crys-talline environment that will concern us here is known as quenching ofthe orbital angular momentum under certain circumstances. This effect is


observed for instance in magnetization measurements. We define the ef-fective magneton number p as the modulus of the ionic magnetic momentin units of Bohr magnetons:

p = g√

J(J + 1)

where J = total angular momentum of ion in units of h and g = gyromag-netic ratio (see Sect. 1.3). One verifies that agreement with experimentsin the estimate of p for the Cr3+ ion is only achieved if the ground stateexpectation value of L is assumed to vanish in the crystal, although for thefree ion L = 3. This quenching of L is the result of the local symmetry ofthe electrostatic potential at the Cr3+ ion site in the solid. This potentialgenerates the so called crystal-field , which, having a symmetry lower thanspherical, in general mixes degenerate atomic orbitals with different ML

values, and lifts the degeneracy of the orbital manifold of states. The neworbital ground state will be the lowest energy one among those arisingfrom the original ground state multiplet under the effect of the crystal-field potential. When the symmetry of the crystal-field admits a singletorbital ground state one can prove that time reversal invariance leads to< ~L >= 0. This theorem is proven in Section 1.5 and Appendix B. Weshall see that a non-degenerate ground state requires an even number ofelectrons in the outer shell of the ion.

In the experiments we refer to in this chapter the Zeeman term in the Hamilto-nian lifts the 2J + 1 degeneracy of the ground state of the ions. For low fieldsthe level separation is proportional to gµBB which is typically of the order of1K, much smaller than the level separation with the excited states, so that toa good approximation we can neglect all excited states and consider, as in Sect.1.3, a problem very similar to Langevin paramagnetism.

Before we discuss the quantum theory of paramagnetism, we shall make a briefreview of the quantum mechanics of atoms.

1.2 Quantum mechanics of atoms

1.2.1 L-S (Russel-Saunders) coupling

Let us write the Hamiltonian for an atom with Z valence electrons (that is, Zelectrons in shells exterior to a filled atomic core of total charge +Z) as:

H = H0 + V1 + V2 (1.7)

where

H0 =

Z∑

i=1

p2i /2m+ Vc(ri) (1.8)

and Vc(r) is a central effective potential, which is usually calculated in theHartree-Fock approximation. The next two terms in Eq. (1.7) are the corrections

1.2. QUANTUM MECHANICS OF ATOMS 5

to this approximate one-electron potential [3]. The second one contains all theCoulomb interaction corrections to the effective self-consistent potential Vc(ri):

V1 =

Z∑

i<j

e2

| ri − rj |−

Z∑

i=1

Ze2

ri− Vc(ri) (1.9)

Finally, V2 contains the magnetic interactions resulting from relativistic effects,of which the dominant ones are the spin-orbit coupling terms.

The relative quantitative importance of V1 and V2 varies along the periodictable. V1 reflects the fluctuations of the exact electrostatic potential relative tothe Hartree–Fock potential. The average fluctuations should be negligible if theeffective one-electron potential had been well chosen. Their root mean squarevalue is roughly proportional to

√Z for large Z [3].

The contribution of the spin-orbit coupling can be estimated through a sim-ple calculation based on the Thomas-Fermi approximation for the many-electronatom. This yields [3] V2 ∝ Z2.

As an immediate consequence, one expects that the spin-orbit contributionsto the energy become comparable to -or even greater than- the Coulomb cor-rections given by Eq. (1.9), only for the heavier atoms. In the case V1 V2,which applies in the transition elements,

H ≈ H0 + V1 (1.10)

which, being spin independent, commutes with S and with Sz. Besides, rigidcoordinate rotations around the nucleus leave the Hamiltonian (1.10) invariant,so it commutes with the total angular momentum L and with Lz. One cantake advantage of the fact that L and S are independently conserved by sepa-rately adding L =

∑Zi li and S =

∑Zi si, and afterwards combining both to

obtain the total angular momentum J = L + S, which commutes with the totalHamiltonian. This is the L-S or Russel-Saunders coupling. The Hamiltonian

HLS = H0 + V1 (1.11)

can then be diagonalized in the many-electron basis of states

| LSMLMS >

where ML and MS are the z components of L and S respectively.The whole subspace spanned by the basis above, with L and S fixed and all

possible values of ML and MS is called a spectroscopic term. This manifold, ofdimension (2L+ 1)(2S + 1) is customarily symbolized by

2S+1(Spectroscopic symbol for L)

where the symbols for L are the same used for the one electron states, in capitalletters. For instance, 1P (S = 0, L = 1), 3D (S = 1, L = 2), etc.


Since V1 commutes with L and S , it is diagonal in the subspace of a giventerm and it does not mix different terms [3]:

〈LSMLMS | V1 | L′ S′M ′LM

′S〉 = δLL′δSS′δMLM ′

LδMSM ′

SVLS

1 (1.12)

When all one electron occupied atomic shell states are specified one generatesa manifold which in general contains several different states. This manifoldis called configuration. Let us look at a simple example and see how we candecompose a given configuration into several terms. For Si (Z = 14) the con-figuration is 1s22s22p63s23p2. The only shell partially occupied is 3p. Let usnow construct the terms arising from the configuration 3p2, disregarding theclosed shells. Two l = 1 electrons can be combined to form states with L = 2, 1or 0. On the other hand, the spins of both electrons can be combined into asinglet (S = 0) or a triplet (S = 1). According to Pauli’s principle the totalwave function must be antisymmetric under electron exchange. Since within theLS coupling scheme we consider a product of an orbital wave function times aspin wave function, they must accordingly have opposite parity under electronexchange.

We need to know the parity of the orbital states. We can construct an orbitalwave function with given L and M for the two-electron system by making anappropriate linear combination of products of one-electron orbital states. Thelinear coefficients are the Clebsch-Gordan coefficients (C-G in the following)[3]:

| l1 l2 LM〉 =∑

m1 m2

| l1 l2m1m2〉〈l1 l2m1m2 | LM〉 (1.13)

where | l1 l2m1m2〉 = | l1m1〉 | l2m2〉 is a direct product of one-electron kets.Under electron exchange the C-G coefficients have the parity of L [3], so thatthe same is true of the resulting wave function of Eq. (1.13). In the case of Siwe are only left with the terms 1S , 3P and 1D, in order to satisfy the totalantisymmetry required by the Pauli principle. These terms are L-S subspaces,each of degeneracy (2S+1)(2L+1) respectively, spanned by the eigenfunctionsof the Hamiltonian HLS . In Si, the 15 degenerate states available for the 3p2

configuration split into three degenerate subspaces of smaller dimension, corre-sponding to the following terms: a singlet 1S, the term 3P of dimension 9 andthe term 1D of dimension 5. The effect of the Coulomb interaction V1 is gen-erally to lift the degeneracy of the configuration, since the expectation valuesof V1, which are degenerate within each term, can in principle be different fordifferent terms. The term degeneracy can further be lifted by the spin-orbitinteraction V2.

1.2.2 Hund’s rules

As to the ordering of the different eigenvalues VLS1 corresponding to the various

terms (Eq. (1.12)), we can determine which is the ground state of the config-uration, that is, the lowest energy term, by applying Hund’s rules [3]. Let us

1.2. QUANTUM MECHANICS OF ATOMS 7

briefly state these rules here. They dictate that, in order to find the groundstate of ion or atom with a given configuration, one must:

1. choose the maximum value of S consistent with the Pauli principle;

2. choose the maximum L consistent with the Pauli principle and rule 1;

3. to choose J :

(a) if the shell is less than half full, choose

J = Jmin = | L− S | ;

(b) if the shell is more than half full, choose

J = Jmax = L+ S .

Rule 1 allows to minimize the intra-atomic Coulomb repulsion among electronsin the same configuration. The second rule is derived as an empirical conclusionfrom numerical calculations. The magnetic (spin-orbit) interactions are takeninto account in rule 3.

Exercise 1.2Show that according to these rules, the ground state of Si should be 3P0, wherethe subindex on the right of the term denomination is by convention the valueof J .

1.2.3 Spin-orbit splitting

Dirac’s relativistic theory of the atom leads to the following spin-orbit correctionto the energy [3]:

V2 ' h2

2m2c2

Z∑

i=1

li · si1

ri

dV (ri)

dri(1.14)

The contributions involving different electrons, proportional, that is, to theoperators li · sj are usually neglected [4]. More concisely:

V2 =

Z∑

i=1

g(ri)li · si (1.15)

We consider now the matrix elements of V2 within the subspace of a given LSterm. All terms in the sum (1.15) must have the same matrix elements, since thewave functions are antisymmetric by exchange of any pair of electrons. Then,

< LSMLMS | V2 | LSM ′LM

′S >=

Z < LSMLMS | g(r1)l1 · s1 | LSM ′LM

′S > (1.16)


Let us look now at a product like lµsν . This operator transforms underthe symmetry group of H, namely the full rotation group in three dimensions,in the same way as the product LµSν . According to Wigner-Eckart theorem(Appendix A), corresponding matrix elements of both operators within an in-variant subspace of fixed L and S are proportional. Besides, the proportionalityconstant is the same within the whole manifold, and it only depends on thevalues of L and S. In particular, for the degenerate manifold α,L, S of alleigenstates of Hamiltonian HLS with eigenvalue α, we have:

< αLSMLMS | V2 | αLSM ′LM

′S >=

A(LSα) < αLSMLMS | L · S | αLSM ′LM

′S > (1.17)

where A(LSα) is a constant for the whole manifold. The eigenvalues of theoperator L · S can be obtained at once, leading to the eigenvalues of V2:

λJα = 1/2Aαh2 [J(J + 1) − L(L+ 1) − S(S + 1)] (1.18)

The degenerate subspace α,L, S splits into the eigenstates of different val-ues of J . Returning to our Si example, we see that for the term 1S, J = 0. Theterm 1D has J = L = 2. Finally the term 3P can yield J = 2, 1 or 0, so that itsplits, according to (1.18) into three different levels 3PJ . Hund’s rules prescribethat the lowest in energy is 3P0.

Exercise 1.3Obtain the spin-orbit level splitting for the various terms of Si 3p2 configuration,and prove that the center of gravity of the levels remains unchanged.

1.3 The quantum theory of paramagnetism

Let us now consider the eigenstates of the atomic Hamiltonian H in Eq. (1.7),which we shall denote by | En J M〉 . Upon application of a uniform staticexternal magnetic field B the perturbing term to be added to the Hamiltonian,neglecting a diamagnetic correction of second order in the field, is the Zeemaninteraction energy

WZ = −µBB · (L + 2S) (1.19)

Here the angular momentum operators are expressed in terms of h, so thatthey are dimensionless, and the Bohr magneton µB = eh/2me has dimensionsof magnetic moment. The factor 2 for spin arises from Dirac’s relativistic theoryof the electron. Each level En has a degeneracy 2J + 1, which will be lifted byWZ . To obtain the displacements of the energy levels within each multiplet ofunperturbed energyEn it is convenient first of all to choose the quantization axisparallel to the direction of the external field. Then, WZ will commute with Jz.We can again make use of the Wigner-Eckart theorem, since any vector operator,

1.3. THE QUANTUM THEORY OF PARAMAGNETISM 9

and in particular L + 2S, transforms like J. Then, their matrix elements areproportional within the level multiplet (see Appendix A):

〈E0JMJ | (L + 2S) | E0JM′J 〉 = g 〈E0JMJ | J | E0JM

′J 〉 (1.20)

where g is the spectroscopic splitting factor, or Lande g factor. We get for the zcomponent:

< E0JMJ | (Lz + 2Sz) | E0JM′J >= gMJ hδMJM ′

J(1.21)

Then

< E0JMJ |WZ | E0JM′J >= −gµBBMJδMJM ′

J(1.22)

so that WZ completely lifts the degeneracy of the atomic levels. When we arewithin the limits of validity of the LS coupling description of the ionic levels(Eq. (1.10)), that is V1 V2, we obtain for g the expression:

g = 1 +J(J + 1) + S(S + 1) − L(L+ 1)

2J(J + 1)(1.23)

Exercise 1.4Prove Eq. (1.23).

Let us now consider the lowest lying Zeeman multiplet in Eq. (1.22), andassume that the spin-orbit splitting is much greater than the Zeeman level sep-aration:

Ah2 gµBB (1.24)

We are now in a situation very similar to that described by the Langevin theoryof paramagnetism, in which the different possible orientations of the spin of anion determine its energy levels in the presence of an extermal magnetic field.The main change is that the quantum theory of the electronic level structurehas led to space quantization of the angular momentum components. For acanonical ensemble of identical ions, each with total angular momentum J , wecan now calculate the partition function resulting from the discrete energy levelsof Eq. (1.22), and then follow the same process as in the classical case in Sect.1.1, to obtain the z component of the magnetization, as:

mz(a) = (N/V )gJµBBJ(a) (1.25)

where a = gJµBβB, N/V is the volume concentration of magnetic ions and theBrillouin function BJ(a) is defined as:

BJ (a) =2J + 1

2Jcoth

(

(2J + 1) a

2J

)

− 1

2Jcoth

( a

2J

)

(1.26)



For a 1 we obtain the zero field static longitudinal susceptibility:

χ =∂mz

∂B|B=0= (N/V )J(J + 1)g2µ2

Bβ/3 (1.27)

which is analogous to the susceptibility obtained from Eq. (1.6). By comparisonof both equations, we obtain the expression

µ2 = J(J + 1)(gµB)2 (1.28)

for the square of the magnetic moment of the ion in terms of its quantized totalangular momentum. The Curie law is still valid in the quantum case, as long asthe assumptions we made to arrive at the above expression for χ are justified.

In Fig. 1.2, as in Fig. 1.1, the values attributed to J are determined byHund’s rules and g is given by Eq. (1.23). The compounds studied in these ex-periments contain the trivalent ions Cr+++, F e+++ or Gd+++, which accordingto Hund’s rules have the ground state terms 4F3/2,

6S5/2 and 8S7/2 respectively.

Exercise 1.6Verify the term assignments for the ground state of Cr+++, F e+++ and Gd+++

mentioned above.

For the rare earths, Brillouin formula gives very good agreement with exper-iment. Measured moments of the rare earths agree very closely with the freeion picture of paramagnetism as presented above, with the exception of Eu3+

and Sm3+. We can understand the anomalies exhibited by these two ions if weconsider that the theory sketched above assumes that the excited states are suf-ficiently separated from the ground state by the spin orbit splittings that theycan be neglected. This is the same as assuming that kBT Ah2, which impliesthat the probability that the excited states are occupied at the temperaturesof interest is negligible. If this were not the case, we must be prepared to finddiscrepancies, which can result in an effective magnetic moment different fromthe one expected for the ground state multiplet. In Eu3+ and Sm3+, Ah2 iscomparable to kBT at room temperature.

In the case of the iron group the situation is completely different. Theagreement with Eq. (1.27) can only be reached if one assumes that L = 0, andaccordingly J = S. This is an example of the phenomenon called quenchingof the orbital angular momentum. We shall see in the next section that this isa result of the symmetry lowering of the ionic one-electron effective potentialdue to the charge distribution around the magnetic ion. In the rare earths, theeffect of the crystal-field is much weaker, because the electronic orbitals of thef shell, which are the unpaired magnetic electron-states, are much more tightlylocalized in space as compared with those of the d shell, and they are screenedby the s, p and d orbitals.

1.4. CRYSTAL-FIELD CORRECTIONS 11

Figure 1.2: Plot of gµB〈Jz〉, in Bohr magnetons per ion vs. B/T for (1) potas-sium chromium alum ( J = S = 3/2 ) , iron ammonium alum ( J = S = 5/2 )and (3) gadolinium sulfate octahydrate ( J = S = 7/2 ). In all cases g = 2. Thenormalizing point is at the highest value of B/T . From Ref. [5].

1.4 Crystal-field corrections

We shall now consider the effect of the electrostatic interactions between theelectrons of the paramagnetic ion and the electric charge distribution of thenearby non-magnetic ions surrounding it, which are called ”ligands”. The latterproduce a resultant effective potential Vcryst that must be added to the atomicHamiltonian of Eq. (1.7).

The concept of crystal-field was advanced by Becquerel [7] and developedoriginally by Bethe [8], Kramers [9], Van Vleck [10] and others. This subjecthas deserved many reviews [11].

The crystal-field potential is generally weaker than the intra-atomic Coulombinteractions, but it may be comparable with the spin-orbit forces. We distinguishaccordingly three cases, classifying the crystal-field as:

• weak, if it is smaller than the spin-orbit interaction;


• intermediate, if it is stronger than the former, but still weaker than theintra-atomic Coulomb electron-electron interactions;

• strong, if it becomes comparable with the intra-atomic interaction.

We do not expect to find the last case to prevail in predominantly ionic com-pounds, since that would imply the presence of covalent bonding of the magneticions with the surrounding ligands. The crystal-field is weak in the rare earthsand actinides, since the 4f and 5f shells are fairly localized near the core andscreened from the ligands by the outer shell electrons, as was mentioned at theend of last section. It is intermediate in the iron-group elements.

1.4.1 Effects of crystal-field symmetry

The details of how the energy levels of the paramagnetic ions will be split bythe crystal-field, depend on the symmetry of the local environment of the ion.As an example, consider the case of a single 3d electron: S = 1/2, L = 2. Then,according to Hund’s rules, J = 3/2 for the ground state multiplet, which is the2D3/2 term. Therefore, this term has a 5−fold orbital degeneracy. Let us remindourselves of the form of these five wave-functions. In cartesian coordinates theycan be written as the product of a radial function f2(r) times the polynomials:X = yz, Y = zx, Z = xy, Φ1 = x2 − y2, and Φ2 = x2 + y2 − 2z2, which arelinearly independent, as can be verified by noting that they are proportional tolinear combinations of the spherical harmonics Y m

2 [15].

Exercise 1.7Prove that the previous five functions are linearly independent.

Let us now consider the symmetry operations of a cubic array of ions. Wecan always place one magnetic ion at the origin of the cartesian coordinate sys-tem, and choose the cartesian axes coincident with the cubic axes of the atomicarray. One can easily see that a reflection on one of the specular symmetryplanes perpendicular to the respective cubic axis, namely one of the transfor-mations x → −x , y → −y , z → −z, leaves the last two functions invariant,while the first three change sign under two of these reflections. On the otherhand, by performing a 2π/3 rotation around a body diagonal axis of the cube,also a symmetry operation, we get cyclic permutations of the coordinates, like(xyz) → (yzx). A permutation is also thereby obtained among the first tripletX,Y, Z, while under the particular permutation above,

Φ1 → Ψ1 = y2 − z2

Φ2 → Ψ2 = y2 + z2 − 2x2

This correspondence can be written as a linear transformation:

Φ1 = −1/2(Ψ1 + Ψ2)


Φ2 = 3/2Ψ1

with the inverse

Ψ1 = 1/2(Φ1 − Φ1)

Ψ2 = 1/2(3Φ1 + Φ2)

The analysis of the effect of all the 48 symmetry operations of the cube(or, which amounts to the same, of the regular octahedron) upon our five di-mensional set of d functions, would generalize what is already suggested by thesymmetry operations considered above, namely that under this group of pointtransformations the first triplet of functions spans a subspace which transformsonto itself. The same is true for the doublet. Both subspaces are therefore closedunder this group. This conclusion can be arrived at in a very straightforwardand elegant way by means of the application of the theory of groups. In thelanguage of this theory, the original 5-dimensional subspace spans a reduciblerepresentation of the cubic point group which decomposes into, or reduces to,two irreducible representations: a three dimensional one, called T2 (or Γ5, de-pending on the notation chosen) and a two dimensional one, called E (or Γ2)(see Appendix A) [15]. Under each of the continuous rotations compatible withthe spherical symmetry of the ionic potencial the original 5 functions obey welldefined transformation laws, which carry each of them into a linear combina-tion of them all. The cube, however, has only a discrete, finite set of symmetryoperations, which are of course contained within the infinite group of rotationswhich leave a sphere invariant. Therefore if a cubic perturbation Vcryst(r) isadded to the spherically symmetric one-electron ion potential for the exampleconsidered above of a 2d state, the five L = 2 states must split into a triplet T2

and a doublet E which as we have seen do not mix any more under the sym-metry operations of the cube. Since both subspaces are now disjoint under thesmaller symmetry group, degeneracy of the eigenvalues of both multiplets (thedoublet and the triplet) is not to be expected. Whenever degeneracy occurs thissituation is called accidental degeneracy.

Within the electrostatic approximation, in which the ligands are substitutedby point charges, a simple calculation indicates which of the split levels has thelowest energy. For the present example of a single d state in a 6 − fold simplecubic (or octahedral) coordination the triplet is the lowest level, while in an8− fold b.c.c. coordination or in the tetrahedral case the opposite is true. Forthe square planar symmetry, which is still lower than the cubic one, the tripletsplits into a doublet plus the singlet Z state. As regards the relative ordering oflevels, determining the position of the Φ1 state level requires some calculation.A practical rule can be stated: the orbitals of the central ion directed along theaxes containing ligand ions will have higher energy than those directed awayfrom the ligands [18].

Bethe [8] and Runciman [12] solved originally the problem of calculating thesplitting of the ionic energy levels in crystal-fields of various symmetries, andfor different values of J , in the case that the spin-orbit splitting is much largerthan the crystal-field effect.


A word of caution is pertinent at this point. One finds in the literature,as already mentioned, different nomenclatures for the various symmetry types.They are denoted Γi by Bethe [8]. Mulliken [19] uses the following nomenclatureinstead: the one dimensional representations with even parity under rotationsaround the principal axis of symmetry are called A, and the odd ones B; theletters E, T and G are used for two, three and four dimensional representationsrespectively; a numerical subscript distinguishes different irreducible represen-tations with the same dimension; subscripts g and u denote even (gerade) andodd (ungerade) representations.

Group theory as applied to Quantum Mechanics establishes that if a Hamil-tonian is invariant under a given symmetry group G, each degenerate multiplet ofeigenstates belonging to a given eigenvalue is a basis for an irreducible represen-tation of G. As shown in the example above, the superposition of a crystal-fieldonto the central (spherically symmetric) one electron effective Hamiltonian, will,as a rule, reduce the dimension of the degenerate multiplet subspace, since thesymmetry point group of a crystal, being a (discrete) subgroup of the full ro-tation group, in general contains subspaces of the multiplet which are invariantunder the smaller group, but not so under the full group. By point group is de-noted one that contains rotations and reflection planes ( eventually multipliedby the inversion) around a fixed point (Appendix A).

From all the foregoing considerations, one concludes that in general thecrystal-field will partially lift the orbital degeneracy of the central ion eigen-states. The resulting multiplets must be characterized by the symmetry typesof the crystal point group. We shall denote a state γ of the Γ representation ofthe point group by | Γγ >. For instance, | T2, X >, | E2, Φ1 >, etc.

Whenever covalent effects can be disregarded, a very useful simplificationfor the discussion of the effects of the crystal-field is its substitution by an elec-trostatic field with the same symmetry. This implies in general the replacementof the ligand ionic charge distribution around each paramagnetic ion by a set ofpoint charges. The corresponding electrostatic potential satisfies Laplace differ-ential equation, and it admits accordingly an expansion in spherical harmonics,in the region around the central ion. This expansion can be used, in conjunc-tion with the Wigner-Eckart theorem, to obtain a widely used representation ofthe crystal-field in terms of angular momentum operators, the Stevens operatorequivalents that we shall discuss in next section.

1.4.2 Stevens operator equivalents

Let us consider the important cases of a ligand field with: a) tetragonal sym-metry (4− fold coordination); b) octahedral symmetry (6− fold coordination)and c) 8 − fold, or bcc coordination. The set of N ligand ions of charges qj

located at points Rj produces at the point r the potential

Vcryst(r) =N∑

j=1

qj| Rj − r | (1.29)


We shall now expand this potential in powers of the components of r, assumingthat | r || Rj |. For most purposes, it is sufficient to expand up to terms of thesixth degree. Let us consider in detail the case of octahedral symmetry, whichcorresponds to a paramagnetic ion with a simple cubic environment. Assumethe coordinates of the surrounding ions are (±a, 0, 0), (0,±a, 0), (0, 0,±a) andall the charges are qj = q.

Exercise 1.8Expand Eq. (1.29) up to sixth order, and obtain [13]:

Vcryst(xyz) =6q

a+

35q

4a5

(

x4 + y4 + z4 − 3

5r4)

−(

21q

2a7

)[

x6 + y6 + z6 +15

4

(

x2y4 + x2z4 + · · ·)

− 15

14r6]

(1.30)

If we adopt spherical coordinates, we must first remind ourselves of thespherical harmonics expansion of the unit point charge potential [20]:

1

| Rj − r | =

∞∑

l=0

rl

Rl+1j

l∑

m=−l

4π

2l + 1Y ∗

lm (θrφr)Ylm

(

θRjφRj

)

(1.31)

It is convenient to change to tesseral, or real, harmonics, defined as:

Zl0 = Y 0l

Zclm =

1√2(Y −m

l + Y ml )

Zslm =

i√2(Y −m

l − Y ml )

with m > 0. The expansion ( 1.30) in tesseral harmonics is then [13]:

Vcryst(r, θ, φ) =∞∑

l=0

rl∑

α

γlαZlα (θ, φ) (1.32)

where

γlα =4π

2l+ 1

N∑

j=1

qjZlα (θjφj)

Rl+1j

(1.33)

In Eq. (1.32) the index α contains the azimuthal quantum number m and theindex c or s defined above.


The above expansion in tesseral harmonics is a very convenient one, sinceby expressing the Zlα in cartesian coordinates, one can obtain an immediatecorrespondence with the Stevens operator equivalents, which are extremely use-ful for evaluating matrix elements of the crystal-field potential. Tables of themost usual tesseral harmonics can be found in the review article by Hutchingsalready cited [13]. To illustrate the use of Eq. (1.32), let us mention that forthe octahedral environment the only non vanishing γ ’s are γ00, γ40, γ60, γ

c44 and

γc64, so that the potential is:

Vcryst(r, θ, φ) =

r4 (γ40Z40 + γc44Z

c44) + r6 (γ60Z60 + γc

64Zc64) (1.34)

or in the notation of Bleaney and Stevens [14]:

Vcryst(r, θ, φ) = D†4[Y

04 +

√

5

14(Y −4

4 + Y 44 )] +

D†6[Y

06 −

√

7

2(Y −4

6 + Y 46 )] (1.35)

where the coefficients are:

D†4 =

7√π

3

q

a3r4 , D†

6 =3

2

√π

13

q

a7r6 .

Other expressions are obtained if the quantization axis is not the z axis [13].The next task is to calculate the matrix elements of the crystal-field potential

operator in the basis of the free paramagnetic ion orbital states belonging to thevarious spectroscopic terms as described in previous sections. In the case of the3d transition elements in ionic compounds, the potential energy of the valenceelectrons associated with the crystal- field, namely the operator

Wc = − | e |N∑

i=1

Vcryst(xi, yi, zi) (1.36)

is larger than the spin-orbit coupling, so that the appropriate free-ion states are| LSMLMS >. In the 4f rare-earth group, they are the states | LS J MJ >,as the spin-orbit coupling is larger than Wc. In either case, the crystal-fieldpotential energy operator must be diagonalized within the adequate basis oforbital states.

Let us now briefly describe the fundamentals of Stevens method of operatorequivalents. It consists essentially in the application of the Wigner-Eckart the-orem to the calculation of matrix elements of the operator which results fromthe expansion of Wc in cartesian coordinates. One starts from the homogeneouspolynomials with the proper symmetry, as in Eq. (1.30) above, and then sub-stitutes the one-electron coordinate operators xi, yi, zi by the correspondingcomponents of J, care taken of the non-commuting properties of the latter.


Whenever it applies, one symmetrizes the corresponding product of differentcomponents of J. Some simple examples are:

∑

i

(3z2i − r2i ) → αJ < r2 >

[

3J2z − J(J + 1)

]

≡ αJ < r2 > O02

∑

i

(x2i − y2

i ) → αJ < r2 >[

J2x − J2

y

]

≡ αJ < r2 > O22

∑

i

(x4i − 6x2

i y2i + y4

i ) →

βJ < r4 >1

2

(

J4+ − J4

−

)

≡ βJ < r4 > O44

A few common low-order operators are:

O02 = 3J2

z − J(J + 1)

O22 =

1

2

(

J2+ − J2

−

)

O04 = 35J4

z − 30J(J + 1)J2z + 25J2

z

−6J(J + 1) + 3J2(J + 1)2

O44 =

1

2

(

J4+ + J4

−

)

O66 =

1

2

(

J6+ + J6

−

)

where z is a crystal axis. The constants αJ for second order operators, βJ for4− th order, γJ for 6− th order, etc., depend on Z,L, S and J , but not on MJ .The actual calculation of the matrix elements can be done, if necessary, for justone of them in each multiplet, and it involves the radial integral

< rn >=

∫

rn+2(f(r))2dr

where f(r) is the electron radial wave function [15]. Since the latter is usuallynot accurately known, these averages are taken as parameters and determinedby fitting experimental results.

Whenever other choices than z are more convenient for the quantization axis,the spin operators must be submitted to the corresponding rotation [13].

Some applications of these methods to typical paramagnetic ions can befound in the book by Al’tschuler and Kozyrev [17]. Effects of covalency be-tween the magnetic ion and the ligands are taken into account in the approachof ligand field theory [16].


1.5 Quenching of L

We are now almost in a position to discuss this effect, which has been alreadymentioned before. We need, however, to digress and discuss the time-reversaltransformation properties of the electronic wave functions involved. The lattercan be constructed as a product of a many electron orbital wave function timesa many electron spinor.

We prove in Appendix B that:

1. Any antilinear operator A satisfies the identity:

(< v | A) | u >= ( < v (| A | u >) )∗

for arbitrary < v | and | u >2. The time reversal operator K0 for the orbital part of the electron wave

function is an antiunitary operator. This means: K0K†0 = 1.

3. If | E > is an eigenstate of the stationary Schrodinger equation, witheigenvalue E, and K is the time reversal operator, then K | E > is alsoan eigenstate, degenerate with | E >.

4. For systems without spin, or as long as we consider only the orbital part ofan electron state, the time reversal operatorK can be taken identical to thecomplex conjugation operator K0 with the properties K0 = K†

0 = K−10 ,

or K20 = 1.

Let us now act with K0 on an orbital state which we assume is not degenerate.In this particular case, the resulting state must be linearly dependent upon theoriginal one, since otherwise the latter would be degenerate. Then,

K0 | n >= c | n > (1.37)

with c = some complex constant. Since K20 = 1, applying K0 again on both

sides of Eq. (1.37) yields:| n >= K0 (c | n >) (1.38)

Since K0 is antilinear,K0c = c∗K0

so from Eq. (1.38) we get

| n >= c∗K0 | n >=| c2 || n >

which implies c = eiφ, with φ real.We consider now a matrix element of L, < n | L | m >, and insert K2

0 insidethe bracket:

< n | L | m >=< n | K0

(

K0LK†0

)

K0 | m > (1.39)

But, from the very definition of time reversal we have:

K0LK†0 = −L (1.40)

1.5. QUENCHING OF L 19

Let us now call | v >=| n >, | u >= LK0 | m > and apply the first property ofantilinear operators enumerated above:

< v | (K0 | u >) = [(< v | K0) | u >]∗

or:< n | (K0LK0 | m >) = [(< n | K0) (LK0 | m >)]

∗

Since we have assumed that < n | and < m | are both non-degenerate, we finallyhave:

< n | L | m >= − < n | L | m >∗ ei(φn−φm) (1.41)

We have < n | L | m >∗=< m | L† | n >, but, since L is an observable, L† = L.In the special case n = m Eq. (1.41) yields

< n | L | n >= 0 (1.42)

This is then the effect of quenching of the orbital angular momentum whichwe already mentioned before. This phenomenon is effective if the crystal-fieldsymmetry is low enough that the ground state of the paramagnetic ion be asinglet.

Let us discuss as an example the quenching of the orbital angular momentumof the Cr++ ion in the salt CrSO4.5H2O, where the ion is at the centre of asquare of water molecules, while two oxygen ions are located exactly above andbelow it, in the apical positions [15]. We shall use the international symbolsfor the point groups throughout. The six oxygens in this arrangement formapproximately a regular octahedron. The crystalline field is the sum of a pre-dominantly cubic component (group m3m) and a smaller tetragonal component(group 4/mmm in international notation) [15]. The tetragonal contribution canbe thought of as analogous to that produced by charges arranged on a tetra-hedron. A small distorsion of the square of water molecules gives rise to anadditional, still smaller, orthorhombic term in the potential, with symmetrygroup mmm , of magnitude similar to the spin-orbit coupling. The Hamilto-nian without the spin-orbit interaction is invariant, in a first approximation,under the rotations and reflections of the space coordinates belonging to thecubic group m3m, and besides, being spin-independent, under any spin rota-tion. The chromous ion has the configuration 3d4, with a ground state term 5D0

(verify this). As we have seen before, the L = 2 orbital wave functions trans-form according to the irreducible representation D(2) under the full rotationgroup, and this representation, which has five dimensions, reduces into the sumof the representations Eg (twofold) and T2g (threefold) of the cubic group, thedoublet having the lowest energy [15]. The tetrahedral distorsion splits furtherthe doublet into two orbital singlets, since the tetragonal group has only onedimensional irreducible representations. They are, in this case, A1g , which isthe identity representation, with symmetry basis z2 and B1g , with symmetrybasis x2 − y2. We shall assume, following Ref. [15] that B1g is the lowest in en-ergy. B1g becomes (i.e., it has the same characters as) Ag of the rhombic groupmmm, and then this is the lowest level when all the crystal-field contributionshave been included.


Let us now consider the spin variables. The spin-orbit interaction is invariantunder all simultaneous spin and coordinate rotations because it is a scalar. Thespin representation is in this case that corresponding to S = 2, which is D(2),but due to the restrictions imposed by the crystal-field point group, the onlysymmetry operations are again those of the group mmm. Therefore, the lowestspin multiplet, which contains 5 spin states, transforms under spin and orbitalrotations as the external product

Ag ×D(2) = Ag +Ag +B1g +B2g +B3g (1.43)

(see appendix A), which are five spin-orbital singlets. Eq. (1.43) states that theexternal product considered of the subspaces which are the basis for represen-tations Ag and D(2) is a reducible representation space of the lower symmetrygroup, which decomposes into the external sum of the irreducible basis sub-spaces of the 5 representations which appear on the r. h. s. of the equation.As we shall see in next section, where we analyze the effect of spin reversalon spin, in this case, since the ionic configuration contains an even number ofelectrons, Kramers theorem allows a singlet ground state. As we proved above,the expectation value of any component of the angular momentum in one or-bital singlet like Ag vanishes, and we have complete quenching, just as in thetransition element compounds we mentioned in the previous section.

1.6 Time reversal and spin

Under time reversal, spin angular momentum reverses sign, just as the orbitalone. For systems with spin we must therefore look for a time reversal operatorK such that

KsK† = −s (1.44)

For one electron, the standard Pauli matrices transform under complex conju-gation as

K0sxK0 = sx

K0szK0 = sz

K0syK0 = −sy (1.45)

The basic commutation relations like

[ x, px ] = ih (1.46)

must change sign under K, so that K must be anti-unitary like K0. We thenintroduce a unitary operator U in spin space:

K = UK0 (1.47)

Then, since K20 = 1, we also have U = KK0 and U † = K0K

†. Under U , whichis an operator on spin space, r and p remain unaltered:

UrU † = KK0rK0K† = KrK† = r

UpU † = KK0pK0K† = −KpK† = p (1.48)

1.6. TIME REVERSAL AND SPIN 21

The Pauli matrices transform accordingly under U as:

U sx U† = −sx

U sy U† = sy

U sz U† = −sz (1.49)

These transformations in spin space can be recognized as the result of a rotationof π around the y axis, which can be represented by the operator

Y = e−iπsy/h (1.50)

Therefore Y and U can only differ by an arbitrary phase factor, which we choose= 1, and consequently

K = Y K0 (1.51)

For one electron, upon expanding the exponential in Eq. (1.50) we find

K = −iσyK0 (1.52)

and for a system with N electrons

K =

N∏

j=1

( − iσy(j) ) K0 (1.53)

The square of this operator is

K2 = (−1)N (1.54)

1.6.1 Kramers degeneracy

We have seen that the (degenerate or not) subspace S(E) belonging to theeigenvalue E is invariant under time reversal when this is a symmetry operationof the Hamiltonian. We must consider separately two cases:

• K2 = +1 (even number of electrons). Here we can choose the phases ofthe eigenstates in such a way that

K0 | n 〉 =| n 〉 , ∀n

and we can use the complete orthonormal set of eigenstates as the basisvectors of the representation. This is called a real basis.

• K2 = −1 (odd number of electrons). A consequence of the negative squareof K is that K† = −K, and since K is anti-unitary we have

〈 n | (K | n 〉) =

〈 n |(

K† | n 〉)

= −〈 n | (K | n 〉) = 0 (1.55)

Then all state vectors can be arranged in degenerate orthogonal pairs,whose orbital parts are mutually complex conjugate. This twofold degen-eracy cannot be lifted by any time-reversal-invariant perturbation.


We have just proved Kramers theorem, which can be stated as:

Theorem 1.1 (Kramers)The energy levels of a system with an odd number of electrons are at least two-fold degenerate in the absence of magnetic fields.

1.7 Effective spin Hamiltonian

We have seen that for HLS , L and S are good quantum numbers. As shown inthe previous section, the crystal-field potential maintains L as a good quantumnumber, but it mixes states with different values of ML according to the pointgroup of the ligand charge distribution. Since the crystal-field potential doesnot depend on spin, an eigenstate of the Hamiltonian :

H0 = HLS + Vcryst (1.56)

can be constructed as the external product of kets

| Ψ >=| LΓ γ >| SMS > (1.57)

where Γ, γ indicate respectively the representation and the particular basis statechosen.

Let us consider a case as the Cr salt of previous section, in which the crystal-field has a sufficiently low symmetry that all degeneracies of the ground stateare lifted, resulting in an orbital singlet which we denote | Γ0 >:

H0 | Γ0 >= E0 | Γ0 > (1.58)

while for the excited states | Γ γ > :

H0 | Γ γ >= EΓ γ | Γ γ > (1.59)

We now add to H0 as perturbation the Zeeman term:

Hz = −µB(L + 2S) ·B

describing the interaction with an external field, and the spin-orbit term:

Hs−o = λ(LS)L · S

Let us now calculate, by the use of perturbation theory, the correction to theground state due to the total perturbation

V = Hz +Hs−o

within the orbital manifold consisting of the ground state and the excited statesdenoted above. In this process, we shall consider the spin operators as c -numbers, since the calculation will be done entirely within the orbital space. Asa result we shall obtain an expression for the perturbed energy of the ground

1.7. EFFECTIVE SPIN HAMILTONIAN 23

state which will depend upon the spin operators, thereby becoming an effectivespin Hamiltonian Heff , which eventually must be diagonalized within the spin2S + 1-dimensional subspace.

By direct application of perturbation theory formalism up to second orderin V , we have:

Heff = H(1) + H(2)

where:

H(1) = −µBB· < Γ0 | L | Γ0 >

−2µBB · S + λS· < Γ0 | L | Γ0 > (1.60)

and:

H(2) = −∑

Γ γ

< Γ0 | V | Γ γ >< Γ γ | V | Γ0 >

(EΓ γ −E0)(1.61)

Exercise 1.9Prove that Heff can be written as:

Heff = −µBBαgαβSβ + λ2SαΛαβSβ − µ2BBαΛαβBβ (1.62)

where the second rank tensors g and Λ are:

Λαβ =∑

Γ γ

< Γ γ | Lα | Γ0 >< Γ0 | Lβ | Γ γ >

(EΓ′γ′ −E0)

(1.63)

gαβ = 2δαβ − Λαβ

Let us now discuss some consequences of Eq. (1.63).

1.7.1 Effective gyromagnetic ratio

There will be some contribution of the orbital angular momentum to g in theground state -although by assumption L was quenched in | Γ0 >- due to theadmixture of the higher energy states. This correction is in general anisotropic,since it reflects the symmetry of the crystal-field. In the case in which thereis one axis of highest symmetry (uniaxial symmetry), which is usually chosenas the z axis, the tensor Λ is diagonal in such a coordinate system, with equaleigenvalues Λ⊥ along the perpendicular directions to z, and in general a differentone Λ‖ along z. According to Eq. (1.63) the same is true for the g tensor.

1.7.2 Single-ion anisotropy energy

In the case that the crystal-field is cubic, the Λ tensor is a scalar. If instead wehave uniaxial symmetry, Eq. (1.62) becomes:


Heff = −g‖µBBzSz − g⊥µB (BxSx +BySy)

+D

[

S2z − S (S + 1)

3

]

+1

3S (S + 1)

(

2Λ⊥ + Λ‖

)

−µ2B

Λ⊥

[

(Bx)2

+ (By)2]

+ Λ‖ (Bz)2

(1.64)

where we introduced the single-ion anisotropy constant D:

D = λ2(

Λ‖ − Λ⊥

)

Observe that the quadratic anisotropy term vanishes identically in the caseS = 1/2. The last term in Eq (1.64) yields at low temperatures a constant con-tribution to the static susceptibility, called Van Vleck susceptibility. The termin Heff quadratic in the magnetic field is much larger in magnitude than thequadratic diamagnetic contribution which we are neglecting.

Exercise 1.10Neglect the spin-orbit term in Eq. (1.64), and assume for simplicity that thesystem is cubic. Prove that if both the thermal energy β−1 and the eigenval-ues of Heff are small compared to the orbital excitation energies appearing inthe denominators of the expression defining Λ in Eq. (1.63), then the staticsusceptibility is

χ = χ0 + 2µ2BΛ (1.65)

where χ0 is the Brillouin susceptibility as calculated before.

As an example of the level splittings of the ground state spin multiplet pro-duced by the anisotropy and the Zeeman terms, let us consider the case of aspin S = 3/2 in a uniaxial crystal-field, with the external magnetic field appliedalong the z axis, for simplicity. Then, the spin dependent part of the effectiveHamiltonian is

Heff = −g‖µBBSz +D

[

S2z − 1

3S (S + 1)

]

which is diagonal in the | S,MS > basis. The eigenvalue structure is depictedin Fig. 1.3.

1.7. EFFECTIVE SPIN HAMILTONIAN 25

Figure 1.3: Zeeman splittings for an S=3/2 atom in a uniaxial crystal-field.

References

1. Langevin, P. (1905) J.de Phys. 4, 678.

2. Langevin, P. (1895) Ann. Chem. Phys. 5, 289.

3. A.Messiah, A. (1960) “Mecanique Quantique”, Dunod, Paris.

4. R.E.Trees, R. E. (1951) Phys. Rev. 82, 683; Marvin, H. H. (1947) Phys.Rev. 71, 102; R.G.Breene, R. G. (1960) Phys. Rev 119, 1615.

5. W.E.Henry, W. E. (1952) Phys. Rev. 88, 559.

6. Kittel, C. (1971) “Introduction to Solid State Physics”, John Wiley &Sons, Inc.

7. Becquerel, J. (1929) Z. Phys. 58, 205.

8. Bethe, H. A. (1929) Ann. Phys. 3, 135.

9. Kramers, H. A. (1930) Proc. Acad. Sci. Am. 33, 959.

10. Van Vleck, J. H. (1952) The Theory of Electric and Magnetic Susceptibil-ities, Oxford University Press.


11. Newman, D. J. and Ng, Betty (1989) Rep. Prog. Phys. 52, 699.

12. Runciman, W. A. (1956) Phil. Mag. 1, 1075.

13. Hutchings, M. T. (1964) Sol. State Phys. 16, 227.

14. Bleaney, B. and Stevens, K. W. (1953) Rep. Prog. Phys. 16, 108.

15. Heine, Volker (1977) “Group Theory in Quantum Mechanics”, PergamonPress, International Series in Natural Philosophy, Vol. 91, Chap. IV.

16. Ballhausen, J. C. (1962) “Introduction to Ligand Field Theory”, Mc Graw-Hill.

17. Al’tshuler, S. A. and Kozyrev, B. M. (1974) “EPR in Compounds of Tran-sition Elements”, Halstead Press, John Wiley & Sons (Keter PublishingHouse, Jerusalem, Ltd).

18. Larsen, E. M. (1965) “Transitional Elements”, W. A. Benjamin, Inc.

19. Mulliken, R. S. (1932) Phys. Rev. 40, 55.

20. Jackson, J. D. (1975) “Classical Electrodynamics”, John Wiley & Sons,Inc.

Chapter 2

Interacting Spins

2.1 Weiss model of ferromagnetism

Before considering any microscopic model for ordered magnetic systems, it is in-structive to review Weiss phenomenological model [1] of ferromagnetism, whichintroduces the concept of a local molecular magnetic field. The very existenceof materials which exhibit spontaneous magnetization in equilibrium demands,of course, some kind of interaction which tends to align the atomic magneticdipolar moments, in such a way that they display a coherent pattern in spacewhich, for a hypothetically infinite sample, can extend indefinitely. The simplestpossible example of a long range ordered system is a single-domain ferromagnet.If we start from the ideas developed in the previous chapter on paramagneticsystems, we should assume that, since spins in a ferromagnet spontaneouslyalign parallel at sufficiently low T , there must exist some local field acting oneach of them (or at least most of them) which, in the absence of an externalfield, can only result from the presence of the other spins. If the system iscompletely homogeneous and has a total spontaneous moment M, we expect auniform magnetization m = M/V , where V is the total volume. Then the natu-ral assumption to make is that the local field acting on each spin is proportionalto m:

Bloc = λm (2.1)

which is Weiss local molecular field, introducing a phenomenological constantλ.

We have hitherto disregarded the magnetic dipolar forces, but they are cer-tainly present in every case, so they would appear as the first candidate for thespin-spin interaction responsible for the magnetic molecular field. Applying tothis case the Lorentz theory of electric dipolar systems, which are completelyanalogous [2] to the magnetic ones, we would find that for a macroscopic systemwith external spherical shape, the magnetic dipolar forces yield, for a uniformmagnetization, a local field

Bloc =4π

3m

27

28 CHAPTER 2. INTERACTING SPINS

The minimum potential energy of a spin J in such a field is:

W = −4π

3mgµBJ ∼ −4π

3(JgµB)2/a3

where a is the average separation between neighbouring spins. Temperature,as we have seen for paramagnets, has the effect of counteracting the aligningtorques produced by the external field. The critical temperature Tc ∼ W/kB

turns out to be in the range of ∼ 10−2K. So, in order to explain critical tem-peratures of 100− 1000K of known ferromagnetic substances we need a factorλ in Eq. (2.1) of the order ∼ 104 − 105 instead of ∼ 4 from the Lorentz theory,which rules out dipole-dipole forces as the source of the spin-spin interactionsin ferromagnets of practical interest. Weiss left λ undetermined, and proceededto apply this idea of a local molecular field to Langevin theory of paramag-netism, adequate for classical magnetic dipolar moments. We shall now profitfrom the quantum theory of paramagnetism developed in the previous chapter,and obtain a version of Weiss theory which incorporates the space quantizationof atomic angular momentum. As we have seen already, the passage to theclassical Langevin limit is simply obtained in the limit J → ∞. Upon appeal toBrillouin’s formula for the paramagnetic magnetization, Eq. (1.25), we add thelocal molecular field to the external one, to follow Weiss, so that we can writefor the average magnetic moment per atom m, the modified Brillouin equation:

m(T,B) = gµBJBJ (βgµBJ(B + λm)) (2.2)

2.1.1 Critical behaviour of Weiss model

We shall now verify that this model exhibits a phase transition. We must finda spontaneous (i.e., in the absence of an external field) finite magnetization atlow T , which must vanish above a given temperature. We take the external fieldB = 0. Let us call x = βgJµBm, and rewrite Eq. (2.2) as:

x = (gJµB)2βBJ (λx) (2.3)

Near the critical temperature where the phase transition occurs, the magneti-zation is small, by hypothesis, so that we can expand the r. h. s. of Eq. (2.3)in a power series in x. It can be proved that for any x,

B′J (x) ≤ B′

J (0) (2.4)

Exercise 2.1Prove Eq. (2.4). Using this property, prove that, whenever

T ≤ Tc =J(J + 1)

3kB(gµB)2λ (2.5)

there are two solutions, ±ms(T ) of Eq. (2.3), while if T > Tc the only solutionis m(T ) = 0.

2.2. MICROSCOPIC BASIS OF MAGNETISM 29

This defines Tc as the critical (Curie) temperature for the transition from fer-romagnetic (at low T < Tc) to the paramagnetic (T > Tc) phase.

We can also calculate the static longitudinal susceptibility, by partial dif-ferentiation of the equation of state, Eq. (2.2) with respect to B.

Exercise 2.2Obtain the static longitudinal susceptibility for T > Tc and B = 0:

χ‖(T, 0) =∂m

∂B=

(gJµB)2βB′J (0)

1 − (gJµB)2βλB′J (0)

=(gJµB)2(J + 1)

3JkB(T − Tc)(2.6)

Equation (2.6) is called the Curie-Weiss law. [3] We can also study the be-haviour of m(T, 0) as a function of T in the neighbourhood of the critical tem-perature, or Curie temperature, Tc. This is easily done by expanding m inpowers of the reduced temperature τ = (Tc − T )/Tc:

Exercise 2.3For B = 0, obtain the expansion

m(T, 0)

gJµB=

kTc

(gJµB)2λ√Aτ1/2 +O(τ3/2) (2.7)

Equation (2.7) is compatible with some characteristics of a second order phasetransition, namely:

• the long range order parameter m, which is finite at low T , tends contin-uously to zero at a finite temperature, in this case the Curie temperature,and in the neighbourhood of Tc, m ∼ τβ , where β is one of the criticalexponents which characterize the phase transitions [4];

• the partial derivative of m with respect to T is singular at the transitiontemperature, which implies that β < 1.

The Weiss model of ferromagnetism yields β = 1/2, which is the typical valuefor this exponent in the mean field approximation.

2.2 Microscopic basis of magnetism

As we have seen in our discussion of the Weiss model of ferromagnetism, thepurely magnetic (dipole-dipole) forces are completely insufficient to provide aquantitative explanation for the high values of the Curie temperatures of therelevant magnetic materials, like Fe,Ni and Co, which are in the range of sev-eral hundred K, corresponding in the energy scale to 0.1 eV , characteristic ofelectronic excitations in atomic systems. The recognition that the electronicinteractions, essentially of electrostatic nature, bring about the phenomena re-lated to collective magnetic behaviour, has been one of the consequences of the


quantum theory of the electronic structure of matter. Heisenberg removed thedifficulties encountered by Weiss. Such large values of the molecular field as areneeded to reproduce the measured Curie-Weiss temperatures, could be justifiedwhen he attributed the interactions between different spins in a magnetic sys-tem to electron exchange effects [7]. The idea, further developed by Dirac [8]and Van Vleck [9], was that the combined effect of the Coulomb interelectronicinteractions and Pauli’s exclusion principle between two atoms with spins S1

and S2 would produce an effective interaction potential of the form

V = −J(R12)S1 · S2 (2.8)

where J(R) is the exchange energy, a function of the interatomic distance R. IfJ is positive in Eq. (2.8), this effective spin-spin interaction provides a modelof ferromagnetic behaviour. We could now postulate this interaction and goonwards to calculate the Weiss field on this basis. However, we prefer to givefirst the derivation of the exchange interaction for a variety of situations, whichwill essentially cover the most important families of insulators with collectivemagnetic behaviour.

Let us discuss first the case of direct exchange. This is a situation in which theground state orbital wave functions of neighbouring atoms have a non negligibleoverlap. There are however very few cases in which direct exchange, as describedbelow, is the basis of ferromagnetism. CrO2 and CrBr3 are examples of thisbehaviour. In the majority of systems which show ordered magnetic behaviour,other exchange mechanisms must be invoked as the basic interactions, and weshall review these alternatives in the following sections. We leave the discussionof itinerant magnetism of transition metals for a later chapter.

2.2.1 The direct exchange interaction

Let us then consider two atoms a and b at a distance R. We shall disregard thestates of the system in which two electrons are occupying orbitals on the samesite, since the large Coulomb repulsion between them in this case will make suchconfigurations very unlikely in the low energy states involved in the magneticinteractions, which, as previously mentioned, are in the 0.1eV range, while theon-site Coulomb repulsion energies are typically of the order of several eV .This is the so called Heitler-London approximation to the electronic structureof molecules. The Hamiltonian we shall consider is:

H = H1 +H2 + e2/R (2.9)

where H1 includes one electron terms, H2 two electron ones and R is the inter-nuclear distance. We are considering monovalent atoms for simplicity, and

H1 =p21

2m+

p22

2m+ va(r1) + vb(r1) + va(r2) + vb(r2) (2.10)

H2 = e2/ | r1 − r2 | (2.11)


where the atomic core potential of atom a is

va(r) = e2/ | r −Ra | (2.12)

with Ra the position vector of atom a, and correspondingly for the atom b.

We recall now briefly the second quantized representation of the many-electron states, since we shall find it convenient to make a slight generalizationof the usual one to include the non-orthogonality of the atomic states which isthe basis of the direct exchange. Let us then consider one-electron spin-orbitals,which are products of an orbital wave function of atom a or b times the corre-sponding spinor. The internal product of the orbital wave functions centeredon different atoms, like

∫

φ∗a(r)φb(r) d3r ≡ Sab ≡< a | b > (2.13)

is in general non vanishing. We shall use sometimes Dirac bras and kets notationto denote the electron states as above. In the following we call Sab ≡ l. Thisoverlap integral in general could be a complex number. The matrix S is hermi-tian, by definition. Let us now consider the representation of the matter fieldoperator associated with the electrons [10] in a non-orthogonal basis. We shallassume that we have a complete, non-orthogonal set of states, which satisfiesthe completeness relation

∑

m,n

φm(x)S−1mn φ

∗n(x′) = δ3(x − x′) (2.14)

where the matrix S is the overlap matrix :

Smn ≡< m | n >

and we can always assume that the original basis states are normalized, so thatthe diagonal elements of S are unity.

Exercise 2.4Consider the linear transformation

| ν >=∑

n

| n >(

S−1/2)

nν(2.15)

where S is the same overlap matrix defined above. Verify that the new set ν isorthonormal. Prove that if one assumes this set is also complete, then Eq. (2.14)follows.

Let us now consider the matter field operator ψσ(x) that annihilates anelectron of spin σ at point x. This operator can be expanded in the originalbasis set as:

ψσ(x) =∑

n,m

φnσ

(

S−1)

nmcmσ (2.16)


with an obvious notation including spin, and where the operator cmσ annihilatesan electron of the corresponding spin in state m and satisfies the anticommuta-tion relations

cnσ , c†mσ′

= Snmδσσ′

cnσ , cmσ′ =

c†nσ , c†mσ′

= 0

Let us review briefly some of the properties of the second quantized fieldoperators. As a consequence of Pauli’s principle they must satisfy the anticom-mutation relations:

ψσ(x) , ψ†σ′ (x

′)

= δσσ′ δ3(x − x′) (2.17)

ψσ(x) , ψσ′(x′) =

ψ†sigma(x) , ψ†

σ′(x′)

= 0 (2.18)

Exercise 2.5Prove that Eq. (2.16) and the anticommutation relations of the c , c† operatorslead to the correct anticommutation relations (2.17) and (2.18) of the field op-erators.

Let us disregard spin for the moment to simplify the notation, and look intoother properties of the field operators. By its very definition, ψ† creates, uponacting on the vacuum state | 0 >, an electron at the point determined by itsargument:

ψ†(x) | 0 >=| x >On the other hand, the one electron state | a > is created from the vacuum as

| a >= c†a | 0 >

Then, consider the internal product

< x | a >=< 0 | ψ(x) c†a | 0 > (2.19)

Substituting the expansion of ψ from (2.16) into ( 2.19) we find:

< x | a >=∑

n,m

φn(x)S−1nm < 0 | cm c†a | 0 > (2.20)

where< 0 | cm c†a | 0 >= Sma (2.21)

So finally we find that< x | a >= φa(x)

is the wave function of state a.We can also evaluate internal products of two electron states. Consider

| ab >≡ c†b c†a | 0 >


and| 12 >≡ ψ†(x2)ψ

†(x1) | 0 >

Exercise 2.6Prove that

< 12 | ab >= φa(1)φb(2) − φa(2)φb(1) (2.22)

which is the as yet un-normalized Slater determinant.

Exercise 2.7Show that

< ab | ab >≡< cacbc†bc

†a | 0 >= 1− | Sab |2 (2.23)

In the Hilbert space of two electron states, the identity operator is:

1 =1

2

∫ ∫

dx1dx2 | 12 >< 12 |

so that∫ ∫

dx1dx2 < ab | 12 >< 12 | ab >= 2 < ab | ab >

Therefore the normalized Slater determinant for orbitals a and b is:

Φ(12) =φa(1)φb(2) − φa(2)φb(1)

√

2 < ab | ab >

We are now ready to study the diatomic molecule in the Heitler-London ap-proximation (HLA). This approximation consists in neglecting completely thestates with double occupancy of one of the atomic orbitals (which of coursecould only occur if both electrons have different spins). This approximationexcludes the ionic states, in which there is charge transfer to one of the atoms,and considers only the covalent configurations, in which electrons are sharedbetween both atoms. The application of this approximation to the Hydrogenmolecule within a variational approach is a standard example in the QuantumMechanics of molecules. An extension to include ionic configurations called theAMO (Alternant Molecular Orbitals) approximation [11] provides a much bet-ter description of the Hydrogen molecule at short distances than the HLA [12],which in general is expected to be reasonable at a typical internuclear separa-tion, but not at very small or very large distances.

At this point we must re-introduce spin. We consider a state

ψ1(12) =1

√

2 < ab | ab >< 12 | a ↑ b ↑> (2.24)


and the time reversed state | ψ4 >, with ↑ changed into ↓ for both electrons,which is automatically orthogonal to the first because so are the spinors.

Their norm squared is

< ψ1 | ψ1 >=< ψ4 | ψ4 >= 1− | l |2

and we denote the corresponding normalized states as | 1 > and | 4 >.We can also consider states with antiparallel spins:

| Φ1 >=| a ↑ b ↓>Φ2 >=| a ↓ b ↑>

These states are not orthogonal:


< Φ1 | Φ2 >= − | Sab |2≡ −l2

The linear combinations Φ1 ± Φ2 are mutually orthogonal, so that the states

| 2 >=1√

2 − 2l2(| Φ1 > + | Φ2 >)

and

| 3 >=1√

2 + 2l2(| Φ1 > − | Φ2 >)

are normalized and mutually orthogonal, besides being obviously orthogonal inspin space to states | 1 > and | 4 >.

Let us denote by Ea and Eb the eigenvalues of the ground states of atoms aand b when at an infinite distance apart:

(

p2

2m+ vα(r)

)

φα(r) = Eαφα(r) (2.25)

where α = a or b.We can now calculate the matrix elements of H within the four dimensional

manifold we are considering. In the first place, since the Hamiltonian is spin-independent, it does not mix states 1, 4 with 2, 3. By the same token, it doesnot connect 1 with 4, so that within the Heitler-London approximation states 1and 4 are eigenstates.

Exercise 2.9Prove that < 1 | H | 1 >=< 4 | H | 4 > and that

< 1 | H | 1 > = Ea +Eb +1

1 − l2(< a | vb | a > + < b | va | b >)

− l∗ < a | va | b > +l < b | vb | a >1− | l |2 +

k − j

1− | l |2 (2.26)


where we use the definitions of the direct (k)and exchange (j)integrals as:

k =

∫

d3r1 d3r2 | φa(1) |2 1

r12| φb(2) |2 (2.27)

and

j =

∫

d3r1 d3r2 φ

∗a(1)φ∗b (2)

1

r12φb(1)φa(2) (2.28)

It is clear that k ≥ 0. One can also prove that j ≥ 0:

Exercise 2.10Express the Coulomb potential in the definition of j above in terms of its Fouriertransform and prove that j ≥ 0.

It is convenient to add together some one-electron and two-electron terms as:

Kab ≡ k +< a | vb | a > + < b | va | b >

1− | l |2 (2.29)

and

Jab ≡ j +l∗ < a | va | b > +l < b | vb | a >

1− | l |2 (2.30)

We can now consider the remaining two dimensional subspace:


< Φ1 | H | Φ1 >= Ea +Eb +Kab =< Φ2 | H | Φ2 > (2.31)

For the non-diagonal elements:


< Φ1 | H | Φ2 >= −(Ea +Eb) | l |2 −Jab (2.32)

where we have defined| l |2 Jab ≡ Jab (2.33)

It is easy to see that the orthonormalized linear combinations | 2 > and| 3 > are eigenstates of H . In fact, one finds that states | 1 >, | 2 > and| 4 > are degenerate, and the corresponding triplet energy Et is (2.31), whilethe remaining singlet eigenvalue Es is

Es = Ea +Eb −Kab

1+ | l |2 − | l |2 Jab

1+ | l |2 (2.34)


We must add to the eigenvalues obtained so far the Coulomb mutual repulsion

energy of both positively charged cores, e2

Rab.

The splitting between the triplet and the singlet is

Et −Es =2 | l |2 (Kab − Jab)

1− | l |2 (2.35)

Exercise 2.13Show that for the triplet states the total spin S = 1, while MS is 1 for | 1 >, 0for | 2 > and −1 for | 4 >, and that for the singlet | 3 >, S = MS = 0;Show also that within the 4-dimensional subspace considered, one can representthe Hamiltonian by an effective spin Hamiltonian

Heff = E0 − Jeff (Rab)S1 · S2 (2.36)

where

E0 =3Et +Es

4

andJeff (Rab) = Es −Et

Heisenberg [7], Dirac [8] and Van Vleck [9] remarked that the Hamiltonian2.36 favours a ferro(antiferro)-magnetic pairing of the electrons if Jeff > 0 (<0) respectively. On the other hand, if Jeff scales with the overlap squared,which is expected to decrease exponentially with the distance between the spins,this would imply that the range of the exchange interaction be limited to asmall number of neighbouring atoms. A spin Hamiltonian with the Heisenbergform (2.36) can in fact be found in situations more general than the ones weassumed before in applying the Heitler-London approach, and as we shall seeits application is fairly wide. As to order of magnitude, the highest criticaltemperatures of ferromagnets are around a few hundreds to around a thousandK, so that the exchange interaction constant Jeff between n. n. atoms shouldbe around 10−2 to 10−1 eV, a relatively small energy in atomic scale, whichmakes its quantitative determination a difficult task.

2.2.2 The superexchange mechanism

The previous results, particularly (2.36) are relevant as long as there is somenon-zero overlap | l | of the ground state wave functions of atoms a and b. Weexpect that the asymptotic dependence of l on Rab be exponential, so that di-rect exchange should not be expected to have a strong influence in substanceswhere the nearest neighbour spins are situated at a distance R which is largecompared to the mean radius of the atomic ground state wave functions of aand b. One could think that direct exchange might be an important ingredientin transition metals, in which the d atomic orbitals would have an appreciableoverlap. However, in metals the Heitler-London approximation, which excludes


interatomic charge transfer, is certainly inappropriate, and we shall see furtheron that for metals other hamiltonians have been proposed than the simple di-rect exchange Heisenberg one, although within certain approximations one canreconcile the Heisenberg Hamiltonian with the magnetism of narrow band met-als. At any rate, one would be led to expect that collective magnetic behaviourin insulating materials could be well described by the Hamiltonian (2.36). Itturns out, however, that many insulating compounds, particularly many oxidesand halogenides, show collective magnetic behaviour, although the magneticions have ground state wave functions fairly localized in space, and are not firstn. n. of each other, so that no appreciable direct overlap is to be expected inthese cases. One typical example is provided by the extensively studied fam-ily of transition metal fluorides MF2, with M = Fe,Co or Mn, in which thetransition metal ions occupy a body centered tetragonal lattice, each metal ionM2+ being surrounded by a distorted octahedron of F−, as shown in Fig. 2.1.The electronic wave functions of the metal cations do not have any overlap sothat direct exchange is excluded, which led Kramers [13] in 1934 to proposethat a strong admixture of the cation and anion wave functions could be in-voked to couple the cations indirectly, thus providing a mechanism for effectiveexchange between the transition metal spins. The mathematical treatment in-volves fourth order perturbation expansion of the total ground state energy.Anderson [14] revised Kramers perturbation approach, and showed that it ispoorly convergent. Besides, several electron transfer processes can be simulta-neously important, and it becomes necessary to have recourse to methods ofligand field theory. Let us now, however, discuss as an introduction to the mainprocesses in action, the simple minded perturbative model of what has becomeknown as superexchange.

The simplest possible system which will exhibit superexchange behaviouris a molecule consisting of one diamagnetic anion surrounded symmetrically bytwo paramagnetic cations, as shown schematically in Fig. 2.2, which displaysboth the ferromagnetic (FM ) and the antiferromagnetic (AFM) configurationsof the electronic spin distribution.

Figure 2.1: Crystal structure of MnF2


Figure 2.2: Electron configurations of cations and anions in the superexchangeprocess

We shall now simplify drastically the atomic hamiltonian, reducing it to theminimum elements which still contain the basic physics we need to describe.

Let us consider the relevant states.

FM configuration

In this case we have one electron occupying the ground orbital state of eachcation, and they both have parallel spins, while two electrons, naturally withopposite spins, occupy the ground orbital state of the ligand atom, which shallbe denoted p, as a reference to its probable s, p character. The correspondingstate is:

| E0 >=| c†b↑ c†p↓ c

†p↑ c

†a↑ | 0〉

and has total spin 1. Let us consider, to simplify, the symmetric case in whichatoms a and b are equal. In the tight-binding approximation, in which oneneglects the effects of overlap, the energy of this state is

E0 = 2Ea + 2Ep + Up

where Up contains the effects of the Coulomb repulsion between both electronsoccupying the ground state orbital, assumed non degenerate, of the ligand,naturally with opposite spins to satisfy Pauli exclusion principle. Due to theCoulomb electron interaction, there is the possibility of electron transfer fromthe ligand anion to the transition cations and viceversa, so one is led to considerstates with double occupation of the cation orbitals, to either side of the anion.We shall assume that the Coulomb repulsion term Ua of the cation, is muchlarger than that of the anion, because the corresponding ground state orbitals


are more localized. Since we need to consider singly occupied states of the anion,we must allow however for doubly occupied cation states. We include then inthe relevant subspace the state

| a >=| c†b↑ c†p↑ c

†a↓ c

†a ↑ | 0 >

and the symmetric one | b >, with two electrons on cation b. These two excitedstates have unperturbed energy E1 = Ep + 3Ea + Ua. The Hamiltonian in thesecond quantized Fock representation that will contain all the terms describedabove is:

H = H0 + V

where

H0 = Ea

∑

σ

(naσ + nbσ) +Ep

∑

σ

npσ +

Ua (na↑na↓ + nb↑nb↓) + Upnp↑np↓ (2.37)

where the number operator nα = c†αcα, and

V = t∑

σ

(

c†aσcpσ + c†bσcpσ

)

+ h.c. (2.38)

is a simplified expression for the terms responsible for the electron transfer be-tween the anion and the cations. One usually assumes that the two-body contri-bution to the hopping potential V can be substituted by a one-body potential inthe Hartree–Fock approximation, and t is the sum of all possible contributionsto the hopping matrix element.

The AFM configuration

In this case, in which, say, ion a has spin up, and b spin down, the unper-turbed energy is the same as for the FM configuration. The excited states | a〉and | b〉 are the same as before, but now there is the possibility of transfer-ring both electrons from the central ligand anion to the cations, which can beaccomplished by the action of the perturbing hopping potential on one of theexcited states, but not directly on the unperturbed ground state. The new stateto consider is

| ab〉 =| c†b↓ c†b↑ c

†a↓ c

†a↑〉

with unperturbed energyEab = 2Ea + 2Ua

Perturbation Calculation

We can now follow the standard perturbation procedure. Let us then substituteV → λV , where λ is a dimensionless small parameter:

H(λ) = H0 + λV


and we look for solutions of the eigenvalue equation

H(λ) | Ψ〉 = E0 | Ψ〉

Now we expand the exact ground state vector as a linear combination of all thestates compatible with each configuration, as discussed above, with coefficientswhich are a series in λ:

| Ψ〉 =| E0〉 +

∞∑

n=1

∑

α

λnan(α) | α〉 (2.39)

and expand the ground state energy E0 in a power series in λ:

E0 = E0 +

∞∑

n=1

λnεn (2.40)


1. ε1 = ε3 = 0 for the AFMand FMconfigurations, while the second orderterms are equal for both.

2. The first different contribution between both configurations is the fourthorder one.

3. The difference of the ground state energies to this order is

∆ = EAFM − EFM =4t4(Ea −Eab)

(Ea −E0)3(Eab −E0)(2.41)

With this result, we can immediately write an effective-exchange spin Hamil-tonian:

Heff = −J Sa · Sb + const. (2.42)

with J = ∆ from (2.41). Under the assumptions we made regarding the un-perturbed energies, we would have in general Eab − E0 > 0 and Ea − Eab ≤ 0,so J < 0, and the superexchange interaction is generally antiferromagnetic.For more detailed information on superexchange including many references seeRefs. [5] and [6].

2.2.3 The RKKY interaction

Some systems have both localized spins and electrons in delocalized, or itin-erant, states. The main examples are the rare-earth metals and the so-calledmagnetic semiconductors. In these systems there is an important contributionto the spin-spin interaction due to the polarization of the itinerant electronsby interaction with the localized spins. Zener [16] propposed a mechanism forpolarizing the s and p conduction electrons in transition metals by exchange


interaction with the d electrons of the paramagnetic ions. This idea was basedon previous work by Frolich and Nabarro [17], in which they proposed a modelof a long-range interaction between nuclear spins. In that model, the hyperfineinteraction between a nucleus and the conduction electrons induces a polariza-tion of the latter, which can in turn polarize another nucleus at a distance R,resulting in an effective exchange interaction between the nuclear spins. This is,accordingly, an effect obtained by considering second order perturbation contri-butions to the ground state energy of the system. The equivalent process, asapplied to localized spins interacting with conduction electrons, was later de-veloped by Ruderman and Kittel [15] and Bloembergen and Rowland [20]. Thisidea was applied by Kasuya [18] to metallic ferromagnetism and by Yosida [19]to alloys of the kind of a transition metal and a diamagnetic one, like Cu−Mn.In this case an indirect interaction is obtained between local electronic spinsinteracting with the Fermi sea of the conduction electrons. We shall derive thisinteraction in Chap. 9. For the time being, we just mention the relevant resultsin connection with the present discussion of exchange.

Suppose that we consider a dilute alloy, in which a small percentage of mag-netic ions have been dissolved in, for instance, a noble metal. Under reasonableassumptions, one can obtain an effective interaction between two spins at a dis-tance R by second order perturbation theory. Let us consider a metal in whichelectrons in the conduction band have an effective-mass m∗. Assume as wellthat the exchange interaction between a conduction electron of spin s and alocalized spin Sn at rn is local, and has the simple form

Hint = Jsds · Snδ3(xel − rn) (2.43)

Then the effective interaction of spins n and m is

Heff = J2sdSn · SmΦ(Rnm) (2.44)

where the range function Φ of the distance between the localized spins is

Φ(R) =m∗µ2

B

h3

1

R4(sin 2kFR− 2kFR cos 2kFR) (2.45)

The indirect exchange mechanism is considered to be responsible for thecolllective magnetic behaviour of the rare earth metals and of the magneticsemiconductors. In these cases, instead of an alloy with a dilute distribution oflocal spins, every atom, or at least every lattice cell, in the case of the magneticsemiconductors, has an unpaired resultant spin arising from electrons in fairlylocalized orbitals, while other electronic states are itinerant, and a local exchangeinteraction between these two families of electrons gives rise, as seen above, to anindirect long-range effective-exchange interaction between different local spins.

References

1. Weiss, P. (1909) J. de Phys. 6, 667.


2. Kittel, C. (1971) “Introduction to Solid State Physics”, John Wiley andSons, Inc., New York.

3. Curie, P. (1895) Ann. Chim. Phys. 5, 289.

4. Stanley, H. Eugene (1987) Oxford University Press, Inc., New York, Ox-ford.

5. Anderson, P. W. (1964) in “Magnetism”, Rado, G. and Suhl, H. (editors),Academic Press, New York, vol. I, Chap.2.

6. J.B.Goodenough, J. B. (1963) “Magnetism and the Chemical Bond”, In-terscience, New York, Chap. 3.

7. Heisenberg, W. (1928) Z. Phys. 49, 619.

8. Dirac, P. A. M. (1929) Proc. Roy. Soc. (London) A123, 714.

9. Van Vleck, J. H. (1934) Phys. Rev. 45, 405.

10. March, N. H., Young, W. H. and Sampanthar, S. (1967) “The many -bodyproblem in Quantum Mechanics”, Cambridge, at the University Press.

11. Lodin, P.-O. (1962) J. Appl. Phys. 33, 251.

12. Chao, K. A., Oliveira, F. A., De Cerqueira, R. O. and Majlis, N. (1977)Int. J. Quant. Chem. XII, 11.

13. Kramers, H. (1934) Physica 1, 182.

14. Anderson, P. W. (1950) Phys. Rev. 79, 350.

15. Ruderman, M. A. and Kittel, C.(1954) Phys. Rev. 96, 99.

16. Zener, C. (1951) Phys. Rev. 87, 440.

17. Frolich, F. and Nabarro, F. R. N. (1940) Proc. Roy. Soc. (London) A175,382.

18. Kasuya, T. (1956) Prog. Theoret. Phys. 16, 45.

19. Yosida, K. (1957) Phys. Rev. 106, 893.

20. Bloembergen, N. and Rowland, T. J. (1955) Phys. Rev. 97, 1679.

Chapter 3

Mean Field Approximation

3.1 Helmholtz free energy

We start by considering an inequality satisfied by the thermodynamic Helmholtzfree energy [1]. If a system is represented by a Hamiltonian H the Helmholtzfree energy F H is a functional of H and a function of the thermodynamicvariables T and V [2]. In the case of magnetic systems we can neglect in generalthe small variations of volume involved, so that we take T and the external fieldB as the variables defining F [3]. For T and B fixed, one can try to develop aperturbative approach to calculate F , similar to the usual scheme for calculatingthe ground state energy E0 = 〈H〉T=0, by writing the identity

H ≡ H0 + (H −H0) (3.1)

where H0 is some Hamiltonian simple enough to be exactly solved. We canexpand F H as a series in the difference H −H0, around F0 = F H0, withthe hope that the correction terms are small, and that H0 has been chosenwisely enough that the main physical properties of the original system are welldescribed by the first few terms in the series. R. E. Peierls obtained a pertur-bative expansion for the free energy which starts like

F = F0 + 〈H −H0〉0 + · · · (3.2)

where F0 = − 1β logZ0, Z0 = Trρ0 , ρ0 = exp−βH0, and we use the notation

〈A〉0 ≡ Tr(ρ0A)/Tr(ρ0) (3.3)

for any operator A. The general result proved in Ref. [1] is that the remainigterms in the series 3.2 add to a non-positive correction (which, eventually, canbe infinite if the perturbation series diverges!) irrespectively of the choice ofH0. Therefore, we have a variational principle at our disposal:

F ≤ F0 + 〈H −H0〉0 ≡ φ (3.4)

43

44 CHAPTER 3. MEAN FIELD APPROXIMATION

One can now obtain variational upper bounds for F by minimizing φ with re-spect to some parameters which are included in the definition of H0. The min-imum of φ gives accordingly the best upper bound of F which can be obtainedwith the particular form chosen for H0. Let us now apply inequality (3.4) tothe Heisenberg Hamiltonian with a Zeeman term:

H = −J∑

<ij>

~Si · ~Sj − γB∑

i

Szi (3.5)

Assume only first nearest neighbour (n. n.) interactions on a spin lattice. Letus choose for H0 the Hamiltonian we know how to diagonalize exactly, namelythe Zeeman paramagnetic one. Following Weiss’ idea of a molecular field, wesubstitute the external field by an effective local field on each atom i,

Beffi = B + Bmol

i (3.6)

In an ordered ferromagnet we assume that the local field is the same on all sitesand that it is parallel to the direction of the external field, which is chosen asthe spin quantization axis. Then,

H0 = −γBeffN∑

i=1

Szi (3.7)

as in Sect. 2.1. Then, just as in Sect. 1.1 we have an unperturbed free energyper spin

f0(β,Beff ) = − 1

βlogZ0(β,B

eff ) (3.8)

where, for S = 1/2,

Z0(β,Beff ) = 2 cosh

(

1

2βγBeff

)

(3.9)

In order to evaluate 〈H〉0 and < H0 >0 we need the average magnetic momentof each site

m ≡ 〈Szi 〉0 =

1

Z0Tr(ρ0S

z) = − 1

γ

(

∂f0(β,B)

∂B

)

B=Beff

(3.10)

where Sz is the corresponding operator of a generic site (since all sites areequivalent). Then we have:

〈H〉0/N = −J2νm2 − γBm (3.11)

where ν is the number of first nearest neighbours of a site. On the other hand,〈H0〉0 involves the unknown Bmol :

〈H0〉0/N = −γ(Bmol +B)m (3.12)

3.1. HELMHOLTZ FREE ENERGY 45

Exercise 3.1Verify Eqs. (3.11) and (3.12).

For S = 1/2 the variational function φ in (3.4) is :

φ = − 1

βlog

(

2 cosh

[

βγ

2

(

Bmol +B)

])

− Jνm2 + γBmolm (3.13)

Both variables Bmol and m are variational parameters. The condition(

∂φ

∂Bmol

)

m

= 0 (3.14)

leads to Eq. (3.10), but we must still impose the second condition for an ex-tremum of φ:

(

∂φ

∂m

)

Bmol

= 0 (3.15)

Exercise 3.2Verify that Eq. (3.15) yields

γBmol = 2Jνm (3.16)

Upon substituting (3.16) into (3.10) we get, for S = 1/2 :

m =1

2tanh

[

β

2(2Jνm+ γB)

]

(3.17)

One can easily solve Eq. (3.17) analytically for B as a function ofm. We see thatthe Weiss molecular field parameter λ can be related to microscopic variables:

λ =2Jν

γ

This solves the puzzle created by the impossibility of explaining the high valuesof typical Curie temperatures of the known ferromagnetic substances within theclassical Weiss-Langevin model of paramagnetism, since the molecular field ina transition metal can be as large as 106 gauss.

Let us now discuss in more detail the consequences of the molecular fieldapproximation for ferromagnetic systems in the S = 1/2 case, with only firstn. n. interactions. The spontaneous behaviour of this system, corresponding tothe case B = 0 is described by the self-consistent solution of (3.17). We shallverify that the parameter

K = βJν

controls the kind of solutions to be found. Let us measure kBT in the naturalunit provided by J . For zero applied field, m = 0 is always a solution, at anytemperature. Besides this paramagnetic solution there are two other non-zeroones, ±ms, where s stands for ‘spontaneous’, iff K > 2. Therefore the critical,or Curie-Weiss temperature

Tc =Jν

2kB


separates the ferromagnetic low T phase from the paramagnetic phase whichexists for T > Tc. For zero external field, as T increases from 0 to Tc, thespontaneous magnetization given by the non trivial solution of (3.17) decreasesfrom m(0) = S to zero, as indicated in Fig. 3.1. The thermodynamic average

Figure 3.1: Mean field results for the spontaneous magnetization of a ferromag-net as a function of T/Tc for several values of S.

Sz ≡ m is the adequate long range order parameter. The continuous vanishingof m and the discontinuity in its first derivative as T Tc are signatures of asecond order phase transition. We already found in Chap. 2 that m(T,B = 0)has precisely these properties, as described in particular, by Eq. (2.7). This is thesame critical behaviour predicted by the Van der Waals theory of the gas-liquidphase transition [4] in which the order parameter is ∆v ≡ vg − vl, the differenceof the specific volumes of the gas and the liquid phase, which plays the role ofms, while the pressure p is the analogue of B. The critical temperature of theVan der Waals fluid corresponds to the Curie temperature of the ferromagnetic-paramagnetic transition [4]. One disturbing feature of the mean field results isthat the critical behaviour is independent on the dimensionality of the system,in the sense that it yields a finite Curie temperature — that is, long range orderat finite temperatures — for any physical dimension of the system. However,we shall see that the isotropic Heisenberg model with short range interactionscannot sustain long range order at any finite temperature for dimension less


than three. This contradiction is a consequence of the absence of fluctuationsin the mean field description. Even the simplest harmonic approximation to thetransverse spin fluctuations in a Heisenberg system with short range interactionsleads to the absence of long range order in less than three dimensions, as we shallfind upon studying the effect of spin waves on the order parameter. In terms ofthe reduced temperature τ = (T − Tc)/Tc we found in Chap. 2 (Eq. (2.7)) thatfor τ 0

ms(τ) = C | τ |β (3.18)

where β is one of the so called critical exponents [5] and C is a constant. Wefound that the mean field approximation yields β = 1/2. Experimental valuesfor β [5] are: 0.32 − 0.36 for the liquid-vapour critical point, and 0.32 − 0.39for the magnetic phase transitions in different systems. For the 3D Heisenbergmodel, numerical calculations yield 0.37±0.04 [9]. Just as in the Van der Waalscase, the present results describe thermodynamic equilibrium states. Within themean field approximation, Eq. (3.17) is the equation of state of the ferromagnet.For τ < 0, i.e in the ferromagnetic phase, we shall now discuss in more detailthe mean field results. In Fig. 3.2 the magnetization, as given by (3.17) isplotted as a function of the dimensionless external field b = γB/J . The examplecorresponds to the coordination number ν = 12 , the number of first n. n. inan f.c.c. lattice. We have taken T/Tc = 1/2 and S = 1/2.

For large | B | the system tends to saturation, that is | m |→ S. FromEq. (3.13) we see that for S = 1/2 we can write the variational approximationto the Helmholtz free energy as:

φ

Jν= −1

2

T

Tclog (2 cosh 2(T/Tc)(m+ b/2)) +m2 (3.19)

Let us start from b > 0, and then imagine that the field is slowly (i.e. reversibly)removed so that the system evolution is a succession of equilibrium states, whichin our case are represented precisely by Fig. 3.2. As b decreases further and itbecomes negative, we find a Van der Waals kind of isotherm, similar to a p(v)isotherm for a gas below Tc. As in that case, we should expect that the sectionBCDEF of the curve be a series of metastable or unstable states. The analogueof the two separate phases of the gas-liquid system is in our case the pair ofdifferent solutions m(b) of the mean field equations for given b. Since Fig. 3.2represents a succession of equilibrium states, then at all points on the curve

φ′ ≡(

∂φ

∂m

)

b

= 0

At an equilibrium configuration, on the other hand, we must have a positivesecond partial derivative: φ′′|b > 0. Upon differentiating Eq. (3.13) twice withrespect to m one obtains:

φ′′|b = −Jν[

βJν(

1− 4m2)

− 2]

(3.20)

which has two roots, ±m0, where m0 depends on the parameters. For positivem we have φ′′|b > (<)0 if m > (<)m0.


Figure 3.2: Equilibrium isotherm of an S = 1/2 f.c.c. ferromagnet for kBT/J =5/2 in the MFA. The average m = 〈Sz〉 is plotted vs. the dimensionless externalfield b = γB/J .

Exercise 3.3Prove that:

• φ as function of m for constant b has local minima (stable configura-tions) on the mutually symmetric portions ABC and EFG of the curve inFig. 3.2, while it has local maxima (unstable configurations) on the portionCDE.

• the partial derivative (∂m/∂b)φ on the curve is infinite at points C andEwhere b = − (+)b0 and b0 depends on the parameters of the system.

Figure 3.3 shows φ(m, b) in a 3D view. Sections of φ(m)b=const. are shown.The Helmholtz free energy obtained in Eq. (3.13) is invariant under the simul-


Figure 3.3: Surface of the variational Helmholtz free-energy φ on the m, b planefor an S = 1/2 f.c.c. ferromagnet.

taneous inversion of m and b. For b = 0 there are two symmetric solutions ±ms

and the two points B and F on the isotherm are symmetric local minima of φ.We choose b < 0 for convenience with the graphical representation. For b < 0points on portion BC of the isotherm in Fig. 3.3 are local minima. The pointson the CD portion are local maxima (unstable configurations). At fixed b if thesystem is on some point on the piece BC it would relax to the lower φ states onportion FG, were it not for the barrier represented by the intermediate maxi-mum along CD, so that we have metastability on BC . As seen in Fig. 3.3 thebarrier disappears for | b |> b0 at point C, where the maximum and minimumcoincide and the second differential of φ vanishes. One would predict a hys-teresis magnetization cycle if the field oscillates slowly in time between positiveand negative values. However, let us remark that a typical barrier height, asthe one for b = 0 is ∆φ ≈ 1.3J , while the thermal energy in this example iskBT ∼ 2.5J . Therefore, one concludes that, had the thermal fluctuations beencorrectly taken into account, they would have destroyed the metastability alongBC and correspondingly the hysteresis, since the system could, through ther-


mal activation, traverse the free energy barrier. This is just one example of thelimitations of the mean field approximation due to its disregard of fluctuations.

Exercise 3.4Sketch the hysteresis loop corresponding to the example considered above.

3.2 Mean field susceptibility

In the paramagnetic regime the zero field susceptibility is isotropic: the longi-tudinal and transverse components are equal. Let us calculate the longitudinalcomponent:

χ =γN

V

(

∂〈Sz〉∂B

)

B=0

(3.21)

From Eq. (3.17),∂〈Sz〉∂B

= SB′S(x)

∂x

∂B(3.22)

wherex = βS( γB + 2Jν〈Sz〉 ) (3.23)

Finally, we find

χ =γN

V

γS2B′S(x)

1 − 2βS2JνB′S(x)

(3.24)

where x is evaluated at B = 0.We shall presently require the limiting expansions of BS(x) for small and

large x:For small x,

BS(x) =S + 1

3S

(

x−Ax3 + O(x5))

(3.25)

where

A =(2S + 1)2 + 1

60S2(3.26)

For large x,BS(x) = 1 − (1/S)e−x/S + O(e−(2x+x/S)) (3.27)

The form (3.24) for χ shows the enhancement of the paramagnetic (Curie)susceptibility due to the interaction among spins. We expect that in the limitB → 0 , x → 0 aswell if T > Tc. We substitute the derivative of (3.25) into(3.24), and we let T decrease from T Tc, looking for the highest temperatureT ( or the smallest β) at which the denominator of (3.24) vanishes - and thesusceptibility diverges in consequence. This determines βc :

2βcJνS(S + 1)/3 = 1 (3.28)

One verifies that for S = 1/2, kBTc = Jν/2 as found before. At low T the systemshould be in the FM phase, and therefore 〈Sz〉 6= 0. We must bear in mind that,

3.2. MEAN FIELD SUSCEPTIBILITY 51

since the Heisenberg Hamiltonian is completely isotropic, any orientation canbe chosen for the ground state magnetization. If one direction in particular isselected, this breaks this continuous symmetry. Upon the application of a smallfield the magnetization will align along the applied field, no matter how smallthat be. We can assume that we apply a field proportional to N−α with α > 0,so that in the thermodynamic limit the Zeeman energy per spin tends to 0 andthe ground state energy in this limit is not affected by this fictitious field. Thenwe obtain at low T a Weiss ground state with all spins aligned in the directionz of the external field.

Let us calculate the longitudinal susceptibility χ‖ in the S = 1/2 case, where

σ ≡ 〈Sz〉/S = tanh(γβB/2 + νβJσ/4) (3.29)

Then

χ‖(T ) =γ2βρ

4(cosh2(x) − Tc/T )(3.30)

where x is the argument of tanh in Eq. (3.29) and ρ = N/V , the spin volume-concentration.

Exercise 3.5Verify Eq. (3.30).

For T → 0 we obtain

χ‖(T ) → γ2ρ

kBTexp ( − νJ/2kBT ) (3.31)

Finally, for T Tc we can expand cosh2(x) in powers of (T −Tc)/Tc to obtain:

χ‖(T ) → γ2TTcβρ/4

(Tc − T )(3T − Tc)' βγ2Tcρ

8(Tc − T )(3.32)

On the other side of Tc, for T Tc, we have

χ‖(T ) → βγ2ρ/4

1 − νβJ/2(3.33)

We conclude that for T ∼ Tc

χ‖(T ) → γ2ρ

4 | T − Tc | (1 − 1

2θ(Tc − T )) (3.34)

where θ(x) = Heaviside step function. Then, in the neighbourhood of Tc

χ−1‖ (T ) = C | T − Tc | (3.35)

with a different coefficient C below and above Tc.


The exponential decrease of χ‖(T ) as T → 0 is not observed experimentally.One important reason for this is that any macroscopic FM is never found ina single domain, uniformly magnetized state at zero applied field. Both re-orientation of domains magnetization and domain-walls displacement changethe bulk magnetic moment. These processes yield a non-zero susceptibilityeven at very small T . Besides, each specimen exhibits a particular behaviour,which depends on the detailed structure of domains when the magnetization isbeing measured, which reflects in a history dependent response.

3.3 Specific heat of ferromagnet

In the mean field approximation each spin i contributes to the total internalenergy with the same average amount

U/N = −γ〈Szi 〉Bmol = −γλm2 (3.36)

The specific heat per atom at zero external field is then

cB = −2γλm(∂m/∂T )|B=0 (3.37)

Let us now obtain for given spin S the limiting forms of the specific heat indifferent temperature ranges. For T → 0, we find for m the form

m|T→0 → S − exp

(

−2JνS

kBT

)

(3.38)

so that

cB =2S2Jν

kBT 2exp

(

−2JνS

kBT

)

(3.39)

which tends exponentially to 0 as T → 0.For T → Tc we need to consider the expansion of BS(x) up to cubic terms.Let us define the reduced temperature τ ≡ T/Tc, and the reduced magnetic

moment σ ≡ 〈Sz〉/S. For B = 0, we have

x =3Sσ

(S + 1)τ

(see Eq. (3.23)). We find, for T Tc

σ = A

(

Tc − T

Tc

)1/2

(3.40)

and

cB(T−c ) = 5/2

(2S + 1)2 − 1

(2S + 1)2 + 1kB (3.41)

where use has been made of the Brillouin function expansion (3.25).

3.3. SPECIFIC HEAT OF FERROMAGNET 53

Figure 3.4: Specific heat per magnetic ion for a Heisenberg ferromagnet, inunits of kB , as obtained in the mean field approximation. The value of spin isindicated on each curve.[6]

Exercise 3.6Verify Eq. (3.41), obtain the classical limit S → ∞ and give an explanation ofthe result in terms of classical kinetic theory.

Above Tc in the MFA one finds cB(T ) ≡ 0, since both the order parameterm = 〈Sz〉 and the molecular field vanish identically. This is physically not satis-factory, since we expect some contribution to the specific heat even at T Tc.The reason for this un-physical result of the MFA lies in the complete neglectof the short range correlations. The only order parameter in the theory, by con-struction, is the macroscopic long-range order parameter σ, which is an averageover the whole (infinite) system. It is completely reasonable to expect that someshort range order survives above the phase transition. The average 〈Sz

i Szi+δ〉

for a small distance δ should not be identically zero in the paramagnetic phase.Some refinements of the MFA incorporate explicitly one or more short rangeorder parameters in the theory from the start, as we shall discuss presently.We found in the MFA a finite discontinuity of the specific heat at the criticaltemperature, as shown in Fig. 3.4. This is a characteristic feature of a secondorder phase transition [7]. In such a process Gibbs’ free energy per spin gF andgP of the ferromagnetic (F) and paramagnetic (P) phases in equilibrium at Tc

satisfy the two conditions:

gF (Tc)|B=0 = gP (Tc)|B=0

∂gF

∂T(Tc)|B=0 =

∂gP

∂T(Tc)|B=0 (3.42)


Figure 3.5: Specific heat per spin of GdCl3 for B = 0, in units of kB , as afunction of T in K.

while the second derivatives of g are different, resulting in a discontinuity of thespecific heat.

For real systems, when the symmetry of the system is known to belong tothe Heisenberg (isotropic) class in 3d, experiments [5] yield a divergence of thespecific heat at the transition point like

c ∼| τ |−α (3.43)

where τ = (T − Tc)/Tc, and

0 < α < 0.11− 0.17 τ > 0

0 < α < 0.13− 019 τ < 0 (3.44)

Experimental results for the ferromagnetic compound GdCl3 [8] are shown inFig. 3.5. Numerical calculations [9] yield for α the values:

3.4. THE OGUCHI METHOD 55

• −0.14± 0.06 for the 3d Heisenberg FM;

• 0 exactly, which in fact corresponds to a logarithmic singularity, for the2d Ising FM, and

• α ≈ 1/8 for the 3d Ising FM.

3.4 The Oguchi method

We shall now consider corrections to the simple MFA by introducing the con-tribution of the short range static correlations. A convenient short range order(SRO) parameter is

∆ ≡ 〈Si · Si+δ〉/S2 (3.45)

for nearest neighbour sites.Within the MFA, since all spins are independent, the average of the product

above factorizes, and

∆MFA = σ2/S2 (3.46)

so that ∆ vanishes above Tc. If instead we explicitly obtain the exact partitionfunction of the isolated pair of spins, the SRO parameter will not vanish abovethe transition. Upon trying to improve the MFA Van Vleck [10] already in 1937used a method which was later developed by Oguchi [11]. Bethe and Peierlsincorporated several improvements to the self-consistent MFA by consideringseveral convenient SRO parameters [12] which are obtained through the exactcalculation of the partition function of a small cluster of spins, and the substi-tution of the rest by a self-consistent mean effective field. Let us briefly discussthe Oguchi method. To this end we shall first solve exactly the Hamiltonian fora pair of n.n. spins:

H ij = −JSi · Sj − γ(Szi + Sz

j )Beff (3.47)

where Beff is in the direction of the magnetization, chosen as z axis. Considerthe sum of both spins

Sp = Si + Sj

with square

(Sp)2 = S′(S′ + 1) (3.48)

where

0 ≤ S′ ≤ 2S

and a total z component Szi + Sz

j with eigenvalues M ′, where −2S ≤M ′ ≤ 2S.The eigenvalues of the pair Hamiltonian (3.47) are

Ep = −J[

S′(S′ + 1)

2− S(S + 1)

]

− γM ′Beff (3.49)


and the SRO parameter is

∆ =〈S′(S′ + 1)〉 − 2S(S + 1)

S2(3.50)

To simplify the notation we define j ≡ βJ and b ≡ γβBeff . The partitionfunction for the two-spin cluster is

Zpair(j, b) =

2S∑

S′=0

S′∑

M ′=−S′

exp [ (j/2)S′(S′ + 1) + bM ′] (3.51)

Exercise 3.7Show that the partition function for the pair of spins with S = 1/2 is

Zpair(j, b) = 1 + ej(2 cosh b+ 1) (3.52)

and obtain the average pair magnetization

〈M ′〉/S =4ej sinh b

1 + ej(2 cosh b+ 1)(3.53)

For consistency, we require that < Si >MFA satisfy the condition

〈M ′〉pair/S = 2〈Szi 〉MFA/S ≡ 2σ (3.54)

To complete the self-consistency conditions we demand that each spin in thepair be acted upon by the same effective molecular field as any other one fromthe rest of the system not in the pair, which we obtained in previous section as

Bmol = 2JνSσ (3.55)

with the only change that now one nearest neighbour has to be excluded sinceits contribution to the interaction has been already included in the pair Hamil-tonian, so that for the spins of the pair the effective field in the present unitsis

b = 2j(ν − 1)Sσ + b0 (3.56)

with b0 = γBβ and B is the external field. We now substitute (3.56) intoEq. (3.53). From Eq. (3.54) we find

σOguchi =2 sinh (b0 + (ν − 1)jσ)

e−j + 1 + 2 cosh (bo + (ν − 1)jσ)(3.57)

The transition temperature can be obtained from (3.57) choosing B = 0 andassuming that σ vanishes at some Tc. The values of Tc obtained from thenumerical solution of the Oguchi equations turn out a few percents smaller

3.4. THE OGUCHI METHOD 57

than the MFA results [12]. One can compare expression (3.57)for b0 = 0 to theone previously obtained within the MFA,

σMFA =sinh (jνσMFA)

1 + cosh (jνσMFA)(3.58)

Exercise 3.8a) Show that one finds again the critical behaviour

σOguchi ∼ (TOguchic − T )1/2 (3.59)

b) Obtain the longitudinal susceptibility below Tc in the Oguchi approximation :

χ = limB→0

Nγσ

V B=

(

C

T

)

4

e−2j + 3 − 2(ν − 1)j(3.60)

where C = Curie constant, and show that it has the same critical behaviour asin the MFA:

χ−1 ∼| Tc − T | (3.61)

We show now that the SRO parameter ∆ defined in (3.45) does not vanishin the Oguchi approximation at any finite T . The cluster (in this case, pair)average we need is

〈S′(S′ + 1)〉 =2(1 + 2 cosh b)

e−j + 1 + 2 cosh b(3.62)


If j > 0 we expect ∆ > 0. We can verify this for T → 0 and for T Tc.In fact, as T → 0, ∆ → 1. In the limit T Tc we get

∆ → 3(1− e−j)

3 + e−j(3.63)

which is positive for j > 0. At the transition temperature ∆ is continuous butits slope is not, due to the discontinuity in the slope of σ. The specific heatper spin at constant B can be calculated from the SRO parameter, since ∆ isproportional to the contribution to the internal energy of each pair of spins inthe absence of an external field:

c = C/N = −JνS2

2

∂∆

∂T(3.64)

For T > Tc, where σ ≡ 0, we obtain

c = 6νkBj2e−2j

(3 + 2e−2j)2(3.65)

This specific heat has a discontinuity at Tc and it does not vanish above Tc, butinstead decreases monotonically. For very large T (very small j) it decreases likeT−2 . This behaviour agrees qualitatively with experiments on FM materials,except very near the Curie temperature.


3.5 Modulated phases

The MFA can yield a great variety of equilibrium phases besides the ferromag-netic one. Suppose that the applied static field oscillates in space. Then theZeeman term becomes

−γB∑

i

Szi cos (q0 ·Ri) (3.66)

We introduce now the Fourier transformed (FT) spin operators

Sq =∑

i

e−iq·RiSi (3.67)

Since we are assuming translational invariance of H , the effective exchangeintegrals Jij depend only on the difference (Ri −Rj), and we obtain for theHamiltonian in wave vector space

H = − 1

N

∑

q

[ J(q)Sq · S−q − 1/2γB(Sz(q) + Sz(−q)) ] (3.68)

We now linearize H to get a mean field approximation equivalent to the staticcase, by use of the identity

Sq ≡ (Sq − 〈Sq〉) + 〈Sq〉 (3.69)

The first term in brackets above is the fluctuation of the spin operator. We nowsubstitute (3.69) into (3.68) and neglect all terms which are of second order inthe fluctuations, thereby obtaining

H ≈ − 1

N

∑

q

[ J(−q) (Sq · 〈S−q〉 + 〈Sq〉 · S−q)−

γB0/2 ( Sz(q0) + Sz(−q0) ) ] (3.70)

If we assume that only the wave vectors ±q0 are important in the response tothe modulated field, then we find an effective molecular field

Beff (q0) = 1/2B +J(−q0)

N〈S(−q0)〉/γ (3.71)

The linearization procedure leads exactly to the equations of the MFA, so it isequivalent to the variational approximation we developed at the beginning ofthis chapter for the Helmholtz free energy. Let us remind ourselves that theresponse function of a non-interacting spin system to a uniform static (that is,ω = 0 , q = 0) external field is Curie’s susceptibility

χ0 = C/T (3.72)

This is the response of independent spins, so if we apply a local external fieldat at lattice site R0, that is if B(R) = Bδ3(R−R0) we must obtain a magne-tization which is also local, of the form M(R) = m0δ

3(R −R0). This impliesthat the site-dependent non-interacting response is

χ0(R) = (C/T ) δ3(R−R0) (3.73)

3.5. MODULATED PHASES 59

whose FT is a constant:χ0(q) = C/T (3.74)

The average magnetization is

M(r) =γ

V

∑

i

δ3(r −Ri)〈Si〉 (3.75)

Therefore the FT of the magnetization for independent spins is aligned in thedirection z of the applied field and its magnitude is

Mz(q) =γ

V〈Sz(q)〉 = χ0(q)Bz(q) (3.76)

We notice that the non vanishing Fourier components of the response (magneti-zation) are only those present in the applied field. Returning now to the case ofinteracting spins, we substitute the effective field for the external one in (3.76),obtaining

〈Sz(q0)〉 =V χ0B0/(2γ)

1 − J(−q0)χ0V/(Nγ2)(3.77)

and consequently

χ(q0) =χ0

1 − J(−q0)χ0V/( Nγ2 )(3.78)

or, substituting χ0 by the expression (3.74),

χ(q0) =C

T − J(−q0)CV/(Nγ2)(3.79)

We consider the case of positive J(R), and a fixed wave vector q0. Supposethat we start from a very large T , where we regain from (3.78) Curie’s law, andstart decreasing T . There will be a value of T for which the denominator in(3.78) vanishes. If the lattice has inversion symmetry, J(q) = J(−q). Then themaximum T for which (3.78) diverges will be proportional to the maximum valueof J(q), so that the critical temperature for the transition from the paramagneticto the ordered phase is

Tc = C(V/Nγ2) max J(q) (3.80)

Exercise 3.10Show that if max J(q) = J(0) we return to the previous results for a FM.

Consider now J(R) < 0, i.e. AFM interactions. In the case that we limitthe range of J to first nearest neighbours, we have

J(q0) = − | J |∑

d

eid ·q0 (3.81)


where d is the set of all translations from a given site to the nearest neigh-bours. If

d · q0 = ±π ∀d (3.82)

then

max J(q) = J(q0) (3.83)

and we find

χ(q0) =C

T− | J | νCV/(Nγ2)(3.84)

for the AFM, so that the Curie and the Neel temperature have the same ex-pression

Tc = TN = CV | J | ν/(Nγ2)

in both cases . We see that for the AFM the longitudinal susceptibility divergesat TN for q = q0, with q0 defined by (3.82), while it remains finite for q = 0.If we depart from the case in which the range of the exchange interaction islimited to the star of first n. n. we find a great variety of phases. For instance,consider a s. c. lattice in which each spin has interactions J1 with its 6 firstn. n. and −J2 with the 12 second n. n., with J1,2 > 0. Then

J(q0) = 2J1 (cos q0xa+ cos q0ya+ cos q0za)

−4J2[(cos q0xa)(cos q0ya) + (cos q0ya)(cos q0za)

+(cos q0za cos q0xa)] (3.85)

Exercise 3.11Show that the extrema of (3.85) satisfy

aq0 =(

cos−1 (J1/4J2))

(1, 1, 1) (3.86)

There are real solutions of Eq. (3.86) if J1 < 4J2. If this is so, in general theparamagnetic phase will be unstable against the spontaneous generation of anon-commensurate magnetization wave, with wave vector as defined by (3.86),at the critical temperature determined by (3.80). The direction of the magne-tization is fixed in this phase. The amplitude of the longitudinal componentvaries periodically as we move in the crystal in the direction of qo, so that thisconfiguration is called a longitudinal (spin) wave (LW).

Exercise 3.12Show that the transition temperature from the paramagnetic to the LW phase is

Tc =S(S + 1)J2

1

4kBJ2(3.87)

3.6. MFA FOR ANTIFERROMAGNETISM 61

and that the uniform susceptibility for this phase is

χ(0)LW =C

T − 8J2(J1 − 2J2)Tc/J21

(3.88)

The coefficient of Tc in the denominator of (3.88) is always < 1, and thisguarantees that the uniform susceptibility is finite at Tc. The high T behaviouris FM like if J1 > 2J2 and AFM like in the opposite case. In the family ofthe rare earths, Tm and Er display this kind of magnetic order within definitetemperature intervals [13].

Static transverse standing waves can also exist. Both transverse and longi-tudinal waves are known as examples of Helimagnetism.

3.6 MFA for antiferromagnetism

As we mentioned before, the AFM instability occurs if Eq. (3.82) is satisfied.Then the average z components nearest neighbour spins tend to have oppositesign. Those lattices that can be subdivided into two interpenetrating sublattices,in such a way that all first n. n. of a site on a given sublattice belong to theother one, are called bipartite. One expects such a lattice with J < 0 to displayAFM order. Going back to Eq. (3.2) we choose as H0 a sum of mean fieldHamiltonians for both sublattices A and B:

H0 = −γ∑

iA

BmoliA

SziA

− γ∑

iB

BmoliB

SziB

(3.89)

where iα are sites on sublattice α, while H is the Heisenberg Hamiltonian ofEq. (3.5) with J < 0. We shall assume that the external field B and the sublat-tice magnetization are parallel to z. In general, there will be some anisotropypresent that will favour the orientation of the magnetization along special crys-tallographic axes, and we assume that z is along one of these axes.

The partition function Z0 for the non-interacting H0 can be obtained imme-diately:


Z0 = Z0(A)Z0(B) (3.90)

where Z0(α) for α = (A,B) can be explicitly written for each sublattice as inSect. 3.1.

For S = 1/2 we already obtained:

Z0(β,B) = 2 cosh

(

1

2βγB

)

(3.91)


Then we write

ZA,B0 = 2 cosh

(

1

2βγBmol

A,B

)

(3.92)

If N is the total number of sites in the whole lattice, we calculate the paramag-netic free energy for each sublattice as

fA,B0 =

FA,B0

N/2= − 1

β(2/N) logZA,B

0 (3.93)

and of course f0 = fA0 + fB

0 . It is convenient to add to the Hamiltonian astaggered field Ba which can be the result of a single site anisotropy term in theenergy or can be considered as an artificially constructed external field, withthe property that Ba(iA)σA > 0 and Ba(iB)σB > 0 where

σA,B = 〈SzA,B〉

Eventually, we shall add a Zeeman term to include a uniform external field B.We now calculate the variational function φ as defined in section 1:

φ = F0 + 〈H −H0〉0 (3.94)

To this end we calculate the averages of H and H0 in the non-interacting en-semble:

〈H〉0N/2

=| J | νσAσB − γ | Ba | (| σA | + | σB |) − γB(σA + σB) (3.95)

and

〈H0〉0N/2

= γBAσA − γBBσB − γ | Ba | (| σA | + | σB |) − γB(σA + σB)

(3.96)

Upon equating to zero the partial derivatives of φ with respect to Bα andσα we find

σA,B = −(1/γ)∂fA,B

0

∂BA,B(3.97)

and

BA,B = − | J | νσB,A (3.98)

We have finally:

σA,B = SBS(SγβBA,B) (3.99)

For S = 1/2 we have

σA,B = (1/2)tanh(βγBA,B/2) (3.100)


Let us now find the critical temperature for S = 1/2. As T TN , σA,B 0and we expand tanh for small argument. Let us call a = βγ | J | ν. The pair ofequations (3.100) reduces to the linear system:

2σA + aσB = 0aσA + 2σB = 0

(3.101)

The eigenvalues of the secular 2× 2 determinant of system (3.101) are ±a, andthe corresponding eigenvectors are

(σA/σB)±

= ∓ 1

where the choice (+) yields an unphysical ferromagnetic order solution, withnegative critical temperature, while the (−) solution corresponds to the AFMorder with antiparallel sublattices and with Neel temperature

TN =ν | J |2kB

(3.102)

which coincides with our previous expression for the ferromagnetic Tc if onemakes the substitution J →| J |.

3.6.1 Longitudinal susceptibility

Equation (3.98) contains implicitly the generalization of Weiss molecular field foranti-ferromagnets. To make this more explicit we introduce as many parametersλα as sublattices. The bipartite AFM with only two Weiss molecular fieldparameters is the next simplest case after the single domain FM:

Bα = −λαβMβ (3.103)

where α, β = A,B. We have

λAB = 2| J | νB

γ2ρA(3.104)

where ρA = ρB = N/(2V ) volume concentration of each sublattice spins andνA,B = number of B(A) spins surrounding an AB spin. The total field on sitesα = (A,B) is

Bα = B − λαβMβ (3.105)

Let us call

bα =γBα

kBT(3.106)

and

b0 =γB

kBT(3.107)

Then,σα ≡ 〈Sz

iα〉 = SBS(S (bα + b0)) (3.108)


and the sublattice magnetization can be written as

Mα = γρσα (3.109)

Let us consider first the paramagnetic phase, so that Mα = χ(α)0 B. Then

bα = b0 − λαβχβ0 b0 (3.110)

We assume both sublattices are atomically identical, so that we have λαβ =λβα = λ and ρα = ρ.Let us now calculate the susceptibility as the response of the total magnetizationM = MA +MB to the external field in the zero field limit, that is

χ =

(

∂M

∂B

)

B=0

(3.111)

Now we expand the Brillouin function in (3.108) for small B, and after somesimple algebra we obtain:

χ‖ =χ0

1 + χ0λ=γ2S(S + 1)ρ

3kB

1

T + θ(3.112)

where

θ =γ2S(S + 1)ρλ

3kB(3.113)

and we substituted in Eq. (3.112) the expression for the paramagnetic suscep-tibility

χ0 =γ2S(S + 1)ρ

3kBT(3.114)

We verify that for S = 1/2 , θ =| J | ν/(2kB) = TN .The curve of χ−1(T ) has the negative intersect −θ with the T axis. In the

special case of only first n. n. interactions θ coincides with TN , but in generalthis is not necessarily the case: if the range of the interaction is greater thanfirst n. n., the ordered phases are more complicated and one may have to dealwith more than two sublattices. Depending on the ratios Jn/J1 one can obtainθ/TN > 1, as found experimentally for many systems.

We remark that if only J1 = − | J |6= 0 , in which case θ = TN = λC/2, weobtain

χ(T )|TTN= 1/λ (3.115)

Now let us turn to the ordered Neel phase at low temperatures with an externalfield B0 applied along the magnetization of the A sublattice. Then for eachsublattice

MA,B = MA,B0 + χ‖B0/2 (3.116)

where MA,B0 is the spontaneous magnetization of each sublattice for B0 = 0.

The response in (3.116) is proportional to χ‖/2, since there are only N/2spins in each sublattice and they occupy the total volume of the system. Call


x0 = SγβλM0 the argument of the Brillouin function in the absence of theexternal field. M0 is the spontaneous sublattice magnetization. One obtains

χ‖ =ρatγ2βS2B′

S(x0)

1 + λρatγ2βS2B′S(x0)

(3.117)

Eq. (3.117) is valid at all temperatures, as one verifies by substituting x0 = 0and obtaining the susceptibility in the paramagnetic phase. We have used2ρA,B = ρat, the atomic volume concentration of the magnetic ions in thelattice.

As T → 0, χ‖ tends exponentially to 0. At T → TN , it has a finite limit,

but the derivative∂χ‖

∂T is finitely discontinuous, due to the discontinuity of thetemperature derivative of the spontaneous sublattice magnetization M0:

Exercise 3.14Show that the derivative of χ‖ with respect to temperature is discontinuous atT = TN .

Figure 3.6: Configuration of magnetization and anisotropy field of both sublat-tices for an applied field B ‖ (z , MA,B) in an AFM.

3.6.2 Transverse susceptibility

Let us find the average energy of the AFM in the presence of an external mag-netic field perpendicular to the spin quantization axis z, which we assume co-


incides with the high symmetry axis of the system. Let us include in the modelHamiltonian an anisotropy energy term of the form

Va = −(1/2)D∑

i,α

(Sziα

)2 (3.118)

so that now the total Hamiltonian H + Va has uniaxial symmetry. In Fig. 3.7we show schematically the situation considered. In equilibrium, we expect bothsublattices to orient symmetrically with respect to the y axis, at an angle φwith the symmetry axis. We neglect in first order the possible change of themagnitude of the sublattice magnetization. We have

〈H0〉0V

= (N/2V )2Jν~σA · ~σB − γ ~B · (~σA + ~σB)

−1/2D[(σzA)2 + (σz

B)2] (3.119)

Figure 3.7: Configuration for transverse field on antiferromagnet.

We are assuming that to first order the average component of spin along theequilibrium orientation of the magnetization is the same, for both sublattices,as it was before the application of the field. Therefore,

~σA(B) = σ0(0, sinφ,+(−) cosφ) (3.120)


Figure 3.8: Longitudinal (‖) and transverse (⊥) static susceptibility of antifer-romagnet for zero field, in MFA. T in units of TN and χ(T ) in units of χ(TN ) .The dotted curve is the expected average susceptibility χP for a powder, in whichall orientations are equally probable. For T > TN , χ is isotropic.

where σ0 is the sublattice magnetization without field as defined in Eq. (3.108)and

〈H0〉0N/2

≡ u(φ) = −2Jνσ20 cos(2φ)

2− γB0σ0 sinφ−Dσ20 cos2(φ) (3.121)

The minimum of u(φ) occurs for

σ0 sin(φ) =γB

4Jν +D(3.122)

which yields for the transverse susceptibility

χ⊥ =1

λ+D/(ργ2)(3.123)

Notice that the temperature dependent σ0 was cancelled out of the expresionabove, so that the transverse susceptibility turns out to be independent on T .The results of the MFA for the longitudinal and the transverse AFM susceptibil-


Figure 3.9: (χ⊥−χ‖ ) per 10−3 mol for MnF2 vs. T in K. The arrow indicatesthe temperature of the maximum in the specific heat [14].

ities are summarized in Fig. 3.8. Experimental values of the difference (χ⊥−χ‖)for MnF2 are shown in Fig. 3.9.

We shall return to the calculation of χ⊥ with Green’s functions methods inChap. 5.

3.6.3 Spin-flop and other transitions

When an external field is applied along the easy axis of an antiferromagnet, theNeel phase can become unstable in relation to the configuration in which thesublattice magnetization aligns approximately perpendicular to the field, whichis why the new phase is called flopped. To verify this let us calculate the MFAenergy in the spin-flopped configuration, represented in Fig. 3.10. Now the fieldB is along the z axis , and we assume that both sublattices align symmetricallyalmost normal to this axis. Then we have (SF stands for spin-flop):

u(φ)SF

N/2= −2Jνσ2

0 − 2γBσ0 sinφ−Dσ20 sin 2φ (3.124)

The minimum of uSF occurs for

σ0 sinφ =γB

4Jν −D(3.125)


Figure 3.10: Spin-flop configuration of antiferromagnet under applied field in aneasy direction.

(compare with Eq. (3.122)). Let us repeat the calculation of the minimumenergy for the AF phase with the applied field parallel to the anisotropy axis andto the magnetization, this time including the anisotrpy energy. We call δσ theinduced change of the sublattice order parameter due to the field, and we assumethat, if the field is applied in the direction of the A sublatice magnetization, thenew equilibrium values are σA(B) = σ0 + (−) δσ. Then we obtain

uAF

N/2= −2Jν[ σ2

0 − (δσ)2 ]

− D

2[ (σ0 + δσ)2 + (σ0 − δσ)2) ] − 2γBδσ (3.126)

Now the minimum uAF occurs for

(2Jν −D)δσ = 2γB (3.127)

and we obtain for the energy

uAF = −2Jνσ20 −Dσ2

0 − (1/2)χ‖B

2

ρ/2(3.128)

We remark here that since we assume thatB and δσ are both positive, Eq. (3.127)shows that there is a maximum anisotropy compatible with the AF phase inthis configuration, namely

D = 2Jν (3.129)


For larger anisotropies the combined anisotropy and Zeeman energies overcomethe exchange energy, and the system aligns with the field, just as in the para-magnetic (P) phase, without undergoing the transition to the flop phase. Amaterial with such a large anisotropy is called a metamagnet. Paradigms ofmetamagnets are FeCl2 and DyPO4.

Let us now study the boundary curves in the (B, T ) plane between differentphases. The difference of the minimum energies of the AF and SF phases is

uAF − uSF = −Dσ2 + (1/2)(χ⊥ − χ‖)B

2

ρ/2(3.130)

Since χ⊥ − χ‖ ≥ 0 , ∃B1 such that for B > B1 the spin-flopped configurationis more stable, and this field is

B1 =

(

Dσ20

1/2(χ⊥ − χ‖

)1/2

(3.131)

Of course the transition only exists for T < TN , because the AF phase dis-appears at that temperature. We see that the critical field B1 for the AF →SF transition depends on the equilibrium magnetization in the absence of field,which in turn depends upon T . Clearly in the absence of anisotropy the criticalfield is zero at any T < TN , so that even an infinitesimal applied field along thespin quantization axis z will produce the spin-flop transition.

In summary, we found that the mean-field energy of the AF and SF phasescoincide when the applied field has the value B1 given by Eq. (3.131). If B > B1

the system will stay in the SF phase. As B increases above B1 we see from(3.125) that there is a second critical field B2 such that

1 = sinφ =γB2

σ0(4Jν −D)(3.132)

For B > B2 both sublattices are aligned parallel to the field, so that this phaseis also called paramagnetic (P): upon increase of T it goes over continuously tothe disordered phase under a field. The anisotropy energies are in most casessmaller than the exchange energies, so in general B2 > B1 at low temperatures.We must still determine the boundary curve AF ↔ P for T < TN , which re-quires to find the value B3 of the applied field such that the energies of the AFand the P phases are equal. In order to obtain the energy for the P phase wemust take into account the change δσ induced by the applied field, which as be-fore can be obtained by minimizing the corresponding energy with respect to δσ.

Exercise 3.16Show that the boundary curve AF ↔ P is

(2Jν +D)σ0 − γB3 = 0 (3.133)


Exercise 3.17Show that if B = B1 = B2, then the same value of B satisfies (3.133) as well.

Figure 3.11: Phase diagram of antiferromagnet in the (B, T ) plane within theMFA.

The conditions B1 = B2 = B3 determine the coordinates of the triple point,at which the three boundary curves meet. These phase boundaries can be cal-culated in the spin wave approximation [15] and with Green’s function methods[16, 17] (See Chap. 5). A very detailed classical reference on the application ofMFA to the phase transitions in antiferromagnets is the review by Nagamiya etal. [18]. Keffer’s [19] monograph on spin waves contains a general review of thesubject.


Phase boundaries obtained in the mean field approximation agree qualita-tively with experimental data on typical AF compounds [20, 21]. We show inFig. 3.12 the experimental results for the phase boundaries of NiCl2.4H2O [21].

Figure 3.12: Phase diagram of NiCl2.4H2O antiferromagnet (from [21]).

3.7 Helimagnetism

Let us consider now the Heisenberg Hamiltonian on a 3d lattice with first andsecond nearest neighbour exchange integrals, and let us assume that the intra-plane interactions are ferromagnetic on all lattice planes belonging to the familyperpendicular to a given direction, say the x axis, so that spins on each plane

3.7. HELIMAGNETISM 73

align parallel at low T . We assume that interactions among different planescould be either positive or negative, so that we expect to find the magnetiza-tion of different planes not necessarily parallel to each other. It is convenientto Fourier analyze the local spin operators. This allows to take advantage oftranslation invariance on the planes. We assume that each plane is ferromag-netically ordered with uniform magnetization. Let us specify a lattice site bythe two component position vector n = (n1, n2) on its plane and the index mof the plane:

H = −∑

m,m′

∑

n,n′

Jm,m′

(n − n′)Sn,mSn′,m′ (3.134)

Here we must include m = m′ in the sum. We assume that the successiveplanes along the x direction are identical 2d lattices, although not necessarily inregistered positions with each other. Now we Fourier transform the local spinoperators:

Sn,m =1√Ns

∑

k

e−ik·nSk,m (3.135)

where k is the 2d wave vector along the plane and Ns = the number of sites oneach plane. The Hamiltonian in terms of the Fourier transformed operators isnow

H = −∑

m,m′

∑

k

Jm,m′

(k) Sk,m · S−k,m′ (3.136)

whereJm,m′(k) =

∑

n

Jm,m′

(n) eik·n

Exercise 3.18Prove that the Fourier transformed operators satisfy the commutation relations

[

Sαk,m , Sβ

k′,m′

]

=1√Ns

i εαβγ δm,m′ Sγk+k′,m (3.137)

Since we have assumed that in the ground state of the system each plane isuniformly magnetized, we can limit ourselves to the k = 0 term in (3.136), andtake

Sk,m = δk,0S0,m (3.138)

We consider the case where spins are aligned parallel to the planes.Our effective Hamiltonian is

Heff = −∑

m,m′

Jm,m′(0) Sk=0,m · Sk=0,m′ (3.139)

whereJm,m′(0) =

∑

h

Jm,m′

(h)

We drop the 0 wave vector index in (3.139) from the FT of the exchange integralsto simplify the notation, and rewrite the effective Hamiltonian as


Heff = −2J1

∑

m

Sm · Sm+1 − 2J2

∑

m

Sm · Sm+2 (3.140)

which can be interpreted as describing a Heisenberg chain. Let us first look atthe semi-classical limit S → ∞. The equilibrium magnetization of each planeis aligned along a given direction, which we take as the quantization axis forspins on that plane, but since we allow the inter-plane exchange interactions tobe either FM or AFM the directions of the magnetization of different planesmight be not parallel, and we should rotate the spin operators in each planeaccordingly so as to make the local quantization axis to coincide with the equi-librium magnetization of the plane. Since all planes are parallel, we can definethis direction for each plane m by specifing θm, the angle the magnetizationat that plane makes with a given fixed z axis. Then the semi-classical energycorresponding to (3.140) is

E

S(S + 1)= −J1

∑

m

cos (θm − θm+1) −

J2

∑

m

cos (θm − θm+2) (3.141)

In an infinite chain all sites are equivalent, so

θm+1 − θm ≡ θ

is m independent, and

θm+2 − θm = 2θ

In Fig. 3.13 we show the assumed spin configuration, where θ is the helixturn angle. If M is the total number of parallel planes (eventually M → ∞) wecan now rewrite the semi-classical energy per plane as

E

S(S + 1)M= −2J1 cos θ − 2J2 cos 2θ (3.142)

the factor 2 on the r.h.s. coming from the neighbours to the left and right of eachplane. Border effects are neglected, and they should be negligible as M → ∞,at least as regards the bulk equilibrium phases. One finds the following extremasolutions:

1. θ = 0 (ferromagnetic order);

2. θ = π (antiferromagnetic order);

3. J1 + 2J2 cos θ = 0 which leads to the helimagnetic ordering.

We call the first two solutions collinear.

3.7. HELIMAGNETISM 75

Figure 3.13: Spin configuration of helimagnet. The lattice constant in the direc-tion of k is denoted c, and the helix turn angle is θ.


∆ ≡ Ecoll −Eheli

8J2=

−S(S + 1)M

4J2

(

8J2 | J1 | +16J22 + J2

1

)

(3.143)

If J2 < 0, the helical phase is stable except at the particular case

| J2 |= (1/4) | J1 | .

If J2 > 0, the collinear phase is stable except at the same special case above,where both types of phases have the same energy. This conclusions do notdepend on the sign of J1, which favours either the F or the AF collinear phase.Helical magnetic (HM) structures were predicted independently by Yoshimori[13], Villain [22] and Kaplan [23], and they were found afterwards in severalmaterials. One can mention the ionic compound MnO2 [13] and the alloyMnAu2 [24] as examples. Also FeCl3 shows a complicated HM structure


[25]. Some heavy rare earths show spiral configurations in some temperatureranges, vith a very rich variety of structures and phase transitions. We refer theinterested reader to Smart’s book [12] for further detail.

3.8 Goldstone’s theorem

We have found within the mean field approximation a low T solution of thevariational equations for the free energy F of a Heisenberg FM which consistsof two pure states, with spontaneous magnetization ±ms. We observe that thesymmetry group of the Hamiltonian is the complete rotation group in 3d, acontinuous orthogonal group with three real parameters (the Euler angles). Inthe Weiss ground state, one particular direction is space has been selected forthe magnetization, so this is a ground state with broken symmetry. Since alldirections in space are equivalent, there is an infinite manifold of states withthe same free energy which are obtained simply by a rigid rotation of all spins,and we find soft modes associated with the transverse fluctuations of long wave-lenght. The reason for this is that there is no energy change associated with aglobal rotation of spins over large distances. This is, for this case, the contentsof Goldstone’s theorem [26], which states that:

If there is a manifold of spontaneously-broken-symmetry degenerate ground-states mutually related by transformations which belong to a continuous group,there are excitations which in the long-wave limit have zero energy.

We check this property in the case of a FM in the ordered state, magnetizedalong the z axis, where the transverse susceptibility χxx is (see Chap. 5):

χxx(ω, k) =γ2ρ

2π

Ek

(ω + iε)2 −E2k

(3.144)

and Ek = 2νJ(1 − γk) + γB. We see that for vanishing ω and B, χxx ∝ k−2.On the other hand, we remind that the longitudinal susceptibility χzz of a FMin the MFA (Sec. 3.) is instead finite in the k → 0 limit for 0 < T < Tc.

References

1. Feynman, R. P. (1972) Statistical Mechanics, W. A. Benjamin, Inc.

2. Sommerfeld, A. (1956) Thermodynamics and Statistical Mechanics, Lec-tures on Theoretical Physics, Vol.V, Academic Press.

3. Stanley, Eugene H. (1971) Introduction to Phase Transitions and CriticalPhenomena, Oxford University Press.

4. Reichl, L. E. (1980) A Modern Course in Statistical Physics, Universityof Texas Press, Austin, Texas, U.S.A.

3.8. GOLDSTONE’S THEOREM 77

5. Patashinskii, A. Z. and Pokrovskii, V. L. (1980) Fluctuation Theory ofPhase Transitions, Pergamon Press, International Series in Natural Phi-losophy, vol. 98.

6. Lidiard, A. B. (1954) Rep. Prog. Phys. 17, 201.

7. Landau, L. D. and Lifshitz, E .M. (1059) Statistical Physics, PergamonPress, London-Paris, p. 434.

8. Garton, G., Leask, M. J., Wolf, W. P. and Wyatt, A. F. G. (1963) J. Appl.Phys. 34, 1083.

9. Ferer, M., Moore, M. A. and Wortis, M. (1971) Phys. Rev.B4, 3954.

10. Van Vleck, J. H. (1937) Phys. Rev.52, 1178.

11. Oguchi, T. (1955) Prog. Theoret. Phys. (Kyoto)13, 148.

12. Smart, J. Samuel (1966) Effective Field Theories of Magnetism ,W.B.SaundersCompany, Philadelphia and London.

13. Yoshimori, A. (1959) J.Phys. Soc.(Japan)14, 807.

14. Griffel, M. and Stout, J. W. (1950) J. Am. Chem. Soc. 72, 4351.

15. Feder, J. and Pytte, E. (1968) Phys. Rev. 168, 640.

16. Anderson, F. B. and Callen, H. B. (1964) Phys. Rev. 136 , A1068.

17. Arruda, A. S. de, et al., (1996) Sol. State. Comm. 97, 329.

18. Nagamiya T., Yosida K. and Kubo, R. (1955) Adv. Phys. 4, 1.

19. Keffer, F. (1966) Handb. der Phys. 2 Aufl., Bd. XVIII Sect. 49,Springer -Verlag.

20. J.N.McElearney, J.N. et al. ,(1973) Phys. Rev. B7, 3314.

21. Paduan Filho, A. (1974) Phys.Lett. 50A, 51.

22. Villain, J. (1959), J. Phys. Chem.Solids 11(1961), 303.

23. Kaplan, T.A. (1960), Phys. Rev. 116, 888.

24. Herpin, A. et al. (1958), Compt. rend. 246, 3170 ; id. (1959), Compt.rend. 249, 1334.

25. Cable, J.W. et al. (1962), Phys. Rev. 127, 714.

26. Goldstone, J. (1961) Nuovo Cimento 19, 154.

Chapter 4

Spin Waves

4.1 Introduction

We have just seen that the Heisenberg Hamiltonian

H = −∑

ij

JijSiSj − γ BN∑

i=1

SZi (4.1)

is a reasonable model for the description of the magnetic properties of sys-tems which contain isolated spins. The derivation of (4.1) was based on theHeitler-London approximation applied to a pair of atoms. We can extend theapplication of that Hamiltonian to multi-electron ions, if we assume that theeffective exchange interactions between any pair of electrons belonging one tothe un-filled shells of ion i, the other to the corresponding shell of j, are thesame for all the possible pairs. Clearly in this case one can add independentlythe spins of each ion, and obtain the Hamiltonian (4.1), where now the spins arethe total spins of the ions, as obtained by applying Hund’s rule to all unpairedspins in the ground state of each ion.

The dimensionless spin operators in (4.1) obey the commutation relations:

[Sαi , S

βj ] = iεαβγ δijS

γi (4.2)

where the indices refer to the cartesian coordinates and εαβγ is the Levi-Civitatensor (totally antisymmetric in the three indices). (4.2) can also be written as

S ∧ S = iS (4.3)

The circular components of S, defined as S± = Sx ± iSy, Sz , satisfy:

[ S+, S− ] = 2Sz

[ S±, Sz ] = ∓S±

79

80 CHAPTER 4. SPIN WAVES

If the spin quantization axis is z,


∑

i

Szi = (Stot)

z

commutes with the Hamiltonian (4.1).

Accordingly the eigenvalue of the z-component of the total angular momen-tum operator is one of the good quantum numbers one can use to label theeigenstates of the Hamiltonian. Of course, if B = 0 , we can choose any com-ponent of Stot as a constant of motion, which is a reflection of the isotropicstructure of Eq. (4.1).

Let us now consider the Hamiltonian (4.1), with B > 0 and assume

Jij ≥ 0 , ∀i, j . (4.4)

Since the electron charge is negative, so is µB , and the minimum potentialenergy configuration in an external field corresponds to the most negative Sz

tot =−NS , N being the number of spins in the system. One can prove that the state| −NS >≡| 0 > is an eigenstate of H, with energy eigenvalue

E0/N = −γB − S2∑

<i,j>

Jij (4.5)

It is easy to see that under the assumption above that all Jij ≥ 0, this statehas the minimum classical energy. This is precisely Weiss ferromagnetic phasefor a single domain sample. Weiss state is also the exact ground state of thequantum Heisenberg ferromagnet if condition (4.4) is satisfied [4].

One obtains N linearly independent states upon introducing in | 0 > onespin deviation at a given site. That is, states in which the spin at site n has its zcomponent increased by one unit: (−S → −S+1), and which have consequentlya total z component of spin −NS + 1 . We shall denote these states | n >.They can be obtained from the Weiss state by acting on it with the local spinraising operator S+

n . We recall the matrix elements of the circular componentsof the angular momentum operator:

S− | S,m > =√

(S −m+ 1)(S +m) | S,m− 1 >

S+ | S,m > =√

(S −m)(S +m+ 1) | S,m+ 1 > (4.6)

Sz | S,m > = m | S,m >

We construct the Weiss state ket forN down spins | 0 > as the direct productof N individual spin kets (spinors), each of which we shall denote by just the meigenvalue since all spinors have spin S:

| 0 >≡N∏

i=1

| −S >i

4.1. INTRODUCTION 81

Then, the ket with one spin deviation with respect to the Weiss state is:

| n >≡| −S + 1 >n

N∏

i6=n

| −S >i

Of course, the quantum Weiss state is a singlet in an applied field, but a differentsituation arises if the external field vanishes, for in that case there is no specialquantization direction, and the Hamiltonian is invariant under the completerotation group in three dimensions, because the exchange energy is a scalar.This implies an infinite degeneracy, as was mentioned in the previous chapter.We can construct orthonormalized states with one spin flip on site n by actingon the Weiss ground state with the S+ operators and using Eqs. (4.6):

| n >=1√2SS+

n | 0 > (4.7)

We assume that the exchange integrals Jij depend only on the difference Rj −Ri. Then, if spins are on sites of a perfect infinite lattice in any dimensions,the Hamiltonian has translational symmetry. One conseqence thereof is thateach exact eigenstate must be a basis for an irreducible representation of thetranslation group of the lattice (appendix A) . This statement is nothing butBloch’s theorem , which in this case establishes that the state

| k >=1√N

∑

n

eik·n | n > (4.8)

transforms as the basis of a symmetry type characterized by the wave vector k.

Exercise 4.2Prove that (4.8) is an eigenstate of H, and that

H | k >= (E0 + εk) | k > (4.9)

withεk = 2S (J(0) − J(k)) + γB (4.10)

where we have defined the Fourier transform of the exchange interaction:

J(k) =∑

h

Jh eik·h (4.11)

and E0 = −2NS2J(0) −NSγB is the ground (Weiss) state energy.

The vectors h above are those connecting each spin to all those interactingwith it through an exchange integral J 6= 0. In the particular case in which onlythe nearest neighbour J is non vanishing, we have, with a simplified notation:

ε(k) = 2SJν(1 − γk) + γB (4.12)


where

γk =1

ν

∑

h

eik·h

is the form factor of the star of first n. n. of each spin and ν = the numberof first n. n. If the lattice has inversion symmetry both εk and γk have thatsymmetry as well.

In the more general case in which the range of J is greater one verifies that aslong as (4.4) is satisfied the magnon dispersion relation (4.10) is positive semi-definite, vanishing at k = 0 in the absence of an external field, in which case thek = 0 mode is a Goldstone boson as mentioned in the previous chapter. Withan applied external field, the excitation spectrum corresponding to exactly onespin deviation has a gap

∆ = γB (4.13)

which is precisely the Larmoor precession frequency. The k = 0 mode is a uni-form (i.e., all spins in phase) precession of the system.

Exercise 4.3Calculate εk of Eq.(4.10) for (a) the linear chain, (b) the square lattice and (c)the three primitive cubic lattices, assuming first n.n. interactions only.

The rotational degeneracy of the isotropic Heisenberg hamiltonian, makes itnecessary, when one considers a single-domain spontaneously magnetized ferro-magnet, to apply an infinitesimal magnetic field, which vanishes in the thermo-dynamic limit at least as N−1+ε, with 0 < ε < 1, such that the energy per spinis not affected in this limit, but the direction of the field defines the spin quanti-zation axis. This field breaks the rotational symmetry of the ground state. TheWeiss state, as defined by this limiting process, is therefore a broken-symmetryground state. In real samples, the rotational symmetry is broken by anisotropicterms which must be added to the Heisenberg isotropic Hamiltonian, like thespin-orbit interaction and the crystal field — sources of single-site anisotropyterms, spin-anisotropy of the exchange interactions and the dipole-dipole inter-action, among others.

4.2 Holstein-Primakoff transformation

Suppose we choose the Weiss ground state with all spins ↑, following the commonuse of disregarding the negative sign of Bohr’s magneton. The applied field isassumed to point also up. Let us now introduce the local spin-deviation operator

n = S − Sz (4.14)

with the eigenvalues

n = S −m

4.2. HOLSTEIN-PRIMAKOFF TRANSFORMATION 83

So, increasing m decreases n and viceversa. For fixed S we label the | m >states in terms of n. For example:

S− | n >=√

(2S − n)(1 + n) | n+ 1 >=

√2S

√

1 − n

2S

√1 + n | n+ 1 > (4.15)

We remind here that the creation operator a† for an energy quantum of a har-monic oscillator satisfies the relation

a† | n >=√n+ 1 | n+ 1 > (4.16)

where n is the number of energy quanta in the state | n >. Therefore wecan rewrite (4.15) in terms of harmonic oscillator creation and annihilationoperators:

S− =√

2Sa†f

S+ =√

2Sfa (4.17)

where the non linear operator f is defined as

f =√

1 − n/2S (4.18)

It is easy to verify that definition (4.17) is compatible with (4.15), that is, solong as

n ≤ 2S (4.19)

The latter is a very stringent condition, although it is easier to satisfy for large S.At any rate this inequality is incompatible with true harmonic oscillators, sincethe spectrum of n is unbounded. We expect however that at low temperaturesthe statistical average of n/2S be small compared to one, so that the squareroot in (4.20) be well defined. Then we follow Holstein and Primakoff (HP) [2]and represent the local spin operators at point i as

S−i =

√2Sa†i fi

S+i =

√2Sfiai (4.20)

Szi = S − ni

Exercise 4.4Prove that if the operators a, a† satisfy boson commutation relations, namely

[ ai, a†j = δij

[ ni, a†j ] = δija

†j

[ ni, aj ] = −δijai

with ni = a†iai, then the transformation (4.20) leaves invariant the commutationrelations of the spin operators.


4.3 Linear spin-wave theory

The operators which create or destroy a state with the required transforma-tion properties under the translation symmetry group of an infinite lattice arethe Fourier transforms of the local spin flip operators, as already indicated byEq. (4.8):

a†k =1√N

∑

n

eik·Rna†n (4.21)

with the corresponding definition for the hermitian conjugate annihilation op-erator. The new harmonic oscillator operators satisfy:

[ ak , a†k′ ] = δkk′

[ ak , ak′ ] = [ a†k , a†k′ ] = 0

[ ak , nk′ ] = akδkk′

[ a†k , nk′ ] = −a†kδkk′

with nk ≡ a†kak We see then that the operators ak, when acting on the vacuumstate | 0 > create the one-deviation states described in Eq. (4.8). These ex-citations are called “ magnons”. The transformation equations of HP, containnonlinear terms which describe interactions between magnons. To see this, letus expand the square root in (4.20) in a power series in n/2S, and let us alsosubstitute on the r. h. s. the local operators by the magnon operators, by usingthe inverse Fourier transform of (4.21). Then we obtain:

S+i =

√

2S

N

∑

k

eik·Riak

− 1

N√

8NS

∑

k k′ k′′

ei(k−k′+k′′)·Ri a†k′ ak′′ak − · · ·

Szi = S − 1

N

∑

k k′

ei(k′−k)·Ri a†kak′ (4.22)

Upon substituting (4.22) for the spin operators in the exchange term of (4.1)one obtains, up to quadratic order in the operators, the harmonic oscillatorHamiltonian:

H(2) = −NνJS2 + 2νJS∑

εka†kak (4.23)

where the dispersion relation εk coincides with that obtained in (4.10). Weshall call the restriction to this Hamiltonian the “free spin wave approximation”(FSWA) for ferromagnets.

For the present choice of the Weiss state with all spins up, the one magnonstates | k > have one quantum less of angular momentum than the Weiss state,i.e. they belong to the subspace of the states with

Sztot = NS − 1

4.4. SEMICLASSICAL PICTURE 85

If we consider a one-magnon eigentate (4.8) we find that the spin deviation isnot localized on any particular site:

Exercise 4.5Calculate the expectation value of the local spin deviation ni in a one magnonstate | k > and prove that

< k | ni | k >=1

N

The exact solution of the Heisenberg Hamiltonian obtained by Bethe [3] in 1dcontains in particular the one-magnon excited states. J. C. Slater [5] proved theexistence of spin-wave-like excitations in a FM ring of N atoms. The applicationof the spin-wave theory to 3D is due to F. Bloch [6] for spin 1/2 and to C. Moller[7] for arbitrary spin. The magnon dispersion relation does not depend uponthe orientation of the wave vector relative to the direction of the magnetization,due to the perfect isotropy assumed for the exchange interactions. In Fig. 4.1we show the instantaneous configurations of spins as a spin-wave propagatesthrough the lattice, in a semiclassical picture in which the spin operators arerepresented as ordinary vectors.

4.4 Semiclassical picture

In the large S limit the quantum description goes over to the classical vectormodel for spin. In this limit, a spin operator can be substituted by an ordinaryvector of length

√

S(S + 1) > S. This can be interpreted as meaning that evenin the Weiss ground state, where for all spins in the lattice Sz = S, there isstill some contribution from the transverse components to S2. The averages< S± >= 0, but the average of the squares of these operators is positive, thuscontributing the remainder of the length of S. In the semi-classical picture,one can think of the spin vectors in the ground state as if they were precessingaround the quantization axis z with random phases. [8]

Klein and Smith [9] show that one can express the difference in energy be-tween the classical and quantum ground state of an FM in terms of the energyof the zero point fluctuations of the quantum oscillators corresponding to theindependent spin waves. To see this, consider the limit of large S. Then thelength of the classical spin vector is S0 ' S + 1/2. We substitute each spin bysuch a vector, and calculate the energy of the Weiss state, including a Zeemaninteraction with an external field B. Then the energy of the classical system is:

Eclass0 = −γBN(S + 1/2)−N(S2 + S)

∑

δ

Jδ (4.24)


Figure 4.1: Pictorial representation of spin wave propagating along a directionperpendicular to the quantization z axis.

Exercise 4.6Prove that the difference between the quantum and classical ground state energiesin the Weiss state is:

Equant0 −Eclass

0 = 1/2∑

k

ε(k) (4.25)

We can also write the classical equations of motion of the angular momentumand verify that the corresponding precession frequency coincides with the spinwave excitation frequency [10]. To this end let us calculate the torque exertedon a magnetic moment m by an effective magnetic field:

T = m ∧Beff = γS ∧ Beff (4.26)

From Euler’s equation of motion of rotating bodies, we can equate the rate ofchange of angular momentum with the applied torque (4.26):

hdS

dt= γS ∧ Beff (4.27)

In the case of a linear chain with only n. n. interactions, the effective fieldacting on spin l is:

Beff = B + (J/γ)(Sl−1 + Sl−1) (4.28)

4.4. SEMICLASSICAL PICTURE 87

so that the equation of motion (4.27) turns into:

hdSl

dt= γSl ∧ [B + (J/γ)(Sl−1 + Sl+1)] (4.29)

We verify that the same equation is obtained quantum mechanically by calcu-lating the commutator of Sl with the Hamiltonian [13]. For the general case ofarbitrary FM exchange interactions in any dimensions, Heisenberg equation ofmotion of the spin operator is

hdSl

dt= hγSl ∧

B + (1/γ)∑

j

Jlj Sj

(4.30)

which reduces to Eq. (4.29) for the chain. We get the conservation law for thez component of the total spin if we sum (4.30) over all sites l. We find that allexchange terms cancel out:


hdStot

dt= Stot ∧ γB (4.31)

where Stot =∑

i Si. Show that this implies

Sztot = const. (4.32)

This result, as we mentioned before (Exercise 4.1), can be directly obtained bycalculating the commutator of Stot with the Hamiltonian. Equations (4.31) and(4.32) can be interpreted as describing a precession movement of Stot aroundthe fixed vector B with the Larmoor frequency ω0 = γB.

Let us now consider solutions of (4.27) which describe a normal mode whereall spins precess at the same frequency. A precession motion with constantfrequency corresponds to a solution of (4.27) in which the transverse componentof S rotates at a constant angular frequency. Then, we look for a solutionsatisfying

dSx

dt= λSy

dSy

dt= −λSx (4.33)

with some real λ. Let us assume that successive spins precess with a phasedifference ka as in Fig. 4.1.

Exercise 4.8

Prove that for the linear chain with only first n.n. interactions,

λ = ω(k) = γB +4J

hS(1 − cos ka) (4.34)

which coincides with the spin wave energy (4.10).


4.5 Macroscopic magnon theory

Let S0 be the spin located at the point chosen as the origin of the coordinatesystem in the lattice. Let δ be the star of its ν first n.n. and assume that Jonly connects S0 with Sδ. We want to analyze the long-wave semi-classicallimit and we substitute the spin operators by c-vectors. In the small k limit thecentral vector and its neighbours in the star considered above are approximatelyparallel, even at high T , in a FM system (J > 0). That is,

| Sδ − S0 | (| Sδ |, | S0 |) .

This assumption is consistent with experimental results on the specific heat(see Chap. 3) and with a vast evidence from inelastic neutron scattering, whichshow the survival of SRO even at elevated temperatures, both in FM and AFMsystems.Then it is natural to expand Sδ in a Taylor series:

Sδ = S0 + (δ · ∇)S0 +1

2(δ · ∇)2S0 + · · · (4.35)

If the crystal has inversion symmetry we find∑

δ

Sδ = νS0 + a2∇2S0 + · · · (4.36)

Expansion (4.35) can be performed for the neighbours of each spin in the crystal,so that the exchange Hamiltonian can be written as

H = −Jν∑

i

S2i − a2J

∑

i

Si · ∇2Si (4.37)

Let us disregard the constant in the r. h. s. above, substitute the sum in thesubsequent term by an integral and neglect derivatives higher than the second,on the basis that the magnetization only varies appreciably over distances muchgreater than a lattice constant. Then

H = −Ja

∫

d3r S(r) · ∇2S(r) + constant (4.38)

We can express the spin field above in terms of the magnetization:

H = − JS2

M20a

∫

d3r M(r) · ∇2M(r) (4.39)

which leads to the definition of the exchange effective field Heff through

H = −1

2

∫

d3r Heff (r) · M(r) (4.40)

where

Heff ≡ 2Js2

M20a

∇2M(r) (4.41)

4.6. THERMAL PROPERTIES 89

and M0 is the saturation magnetization at T = 0.In order to obtain the total effective field we add to (4.41) the external

applied field, the demagnetization field, and eventually the anisotropy field. Weshall have occasion to use this form of the exchange effective field further on.

4.6 Thermal properties

4.6.1 Total spin deviation

Let us now calculate some low T properties of the Heisenberg ferromagnet withinthe FSWA in which magnons act as independent quantum harmonic oscillators.Then an arbitrary excited state is approximately described as a linear super-position of many one-magnon states. This picture makes sense insofar as onecan neglect the effect of the interaction of two or more such excitations . Weshall treat states of two magnons in Chap. 14. It seems in order, however, toremark that in the 3D case, the correction to the energy of a pair of indepen-dent magnons is very small, O(1/N). Bound states of two magnons do exist,but they require a threshold of the single magnon energies which is of the orderof the magnon band-width, and as a consequence of no importance except atvery high temperatures.

The FSWA then leads us to consider an ensemble of boson-type excitationswhich should accordingly obey Bose-Einstein statistics. Let us start the dis-cussion of the effects of temperature by calculating the total spin deviation.Since

Szm = S − nm

at site m, we can obtain the magnetization at a temperature T as:

VM(T ) = γNS − γ∑

m

〈nm〉T (4.42)

where N is the total number of spins and V is the volume of the whole system.

Exercise 4.9Verify that the transformation from the site operators nm to spin wave operatorsnk is unitary and that, accordingly,

∑

m

nm =∑

k

nk (4.43)

Substituting (4.43) into (4.42) we find:

hγNS − VM(T ) ≡ V∆M(T ) = hγ∑

k

〈 nk 〉T (4.44)

Eq. (4.44) gives the decrease of the thermodynamic average of the total magneticmoment of the sample at temperature T from the T = 0 saturated value hγNS.


Let us now consider the thermodynamic limit of (4.44), that is (V,N) → ∞,with limN,V →∞N/V = ρ, the volume concentration of spins in the sample.The summation in (4.44) must be limited to the interior of the first Brillouinzone (BZ). We consider now that the magnons are running waves with periodicboundary conditions in the volume V . In this case the density of points in kspace is V/(2π)d, where V is the total volume and d the dimension of the lattice.The k-space summation can be substituted, in the thermodynamic limit, by anintegration over the volume of the first BZ in d-dimensions:

1

N

∑

k

· · · =v(d)0

(2π)d

∫

ddk · · ·

where v(d)0 is the volume of the lattice unit cell in d dimensions. For a hypercubic

lattice the dispersion relation behaves, in the small k limit, as

εk ' ε0 +Dk2 (4.45)

with a parameter D , which is called the spin stiffness, adequate for each case.

Exercise 4.10Prove (4.45) and obtain the parameter D for the lattices considered in Exercise4.3.

The statistical average of the number of excited quanta of a harmonic os-cillator is the Bose-Einstein probability distribution, so that the total numberof spin deviations at temperature T in thermodynamic equilibrium for the ap-proximation of free spin-waves, is:

∆N =v(d)0

(2π)d

∫

BZ

ddk

eβεk − 1(4.46)

Let us first consider the zero field case: ε0 = 0. At low T, we shall find withreasonable probability only excitations of small energy, of the order of kBT ,and this corresponds to small k, so that the quadratic approximation (4.45) isapplicable. Then the integral in (4.46) is approximately

∫

ddk

eβεk − 1' (βD)−d/2Ωd

∫ ∞

0

ξ(d−2)/2 dξ

eξ − 1(4.47)

where the constant Ωd is the solid angle subtended by the sphere in d dimensions.We find that the number of spin deviations at any non-zero temperature divergesif d ≤ 2. This is the consequence of Mermin-Wagner theorem [1]. For morethan two dimensions this integral can be evaluated in terms of the Riemannzeta function, defined as:

ζ(s) =1

Γ(s)

∫ ∞

0

xs−1dx

ex − 1=

∞∑

p=1

1

ps(4.48)

4.6. THERMAL PROPERTIES 91

For d = 3 we obtain:

∆M(T ) =(

v0/8π3)

4π(βD)−3/2Γ(3/2)ζ(3/2) (4.49)

Then, we have for the magnetization of a simple cubic magnetic lattice withlattice constant a0 :

M0 −M(T ) = γN

V

Γ(3/2)ζ(3/2)

2π2

(

kBTa20

D

)3/2

(4.50)

and this is Bloch’s T 3/2 law [6]. This law is valid, with suitable definitions ofD and a0, for any lattice, and even for a continuous, structureless magneticmedium [8]. Another way of writing (4.50) is:

M(T )/M0 ' 1 − ζ(3/2)Θ3/2 (4.51)

The specific heat per unit volume at zero external field B and constant volumeis given by:

cV =1

V

(

∂E

∂T

)

V

=15M0kB

4γζ(3/2) Θ3/2 (4.52)

In (4.51) and (4.52) Θ is the dimensionless parameter

Θ =

(

γ

M0

)3/2kBT

4πD(4.53)

Exercise 4.11Prove (4.52).

The internal energy of the ensemble of magnons in equilibrium, referredto the ground state energy is

E(T ) =< H >T − < H >0= const. · T 5/2 (4.54)

4.6.2 Non-linear corrections

Let us write the Hamiltonian as

H = −∑

<i,j>

Ji,j(S−i S

+j + Sz

i Szj ) (4.55)

and substitute the local spin operators, according to Eq. (4.22), in terms of theboson operators a, a† , in such a way that the latter appear in normal order (allcreation operators on the left). Then we recover the quadratic Hamiltonian,and we find higher order corrections. If we collect all quartic terms we find thecorrection

H(4) =1

2N

∑

k,k′ ,k′′

(J(k) + J(k + k′ + k′′) − 2J(k − k′′)) ×

a†ka†k′ak′′ak+k′−k′′ (4.56)

as the first higher order correction which has a non-vanishing average.



One can extract from this expression a quadratic form by substituting all pairsof operators of the form a†a by their thermal average with the density matrixcorresponding to the unperturbed harmonic Hamiltonian (4.23), which leads tothe correction

∆H(2) =∑

k

∆ωka†kak (4.57)

equivalent to a correction of the dispersion relation:

∆ωk = − 2

N

∑

q

(Jk + Jq − Jk−q − J0) nq (4.58)

and nq is the Bose-Einstein distribution function. If we restrict the exchangeinteractions to first n.n. range, we can always write Jk = Jγk as in (4.12). Ifwe also assume that the lattice has inversion symmetry, then it is possible toshow that

γk+q = γkγq + · · · (4.59)

where the terms represented by · · · above are odd under inversion. Then

Exercise 4.13Show that the correction to the dispersion relation is

∆ωk = − 1

SNωkE(T ) (4.60)

where E(T ) = total magnon contribution to the internal energy.

According to Eq. (4.54) we get

∆ωk

ωk

∝ (T/Tc)5/2 (4.61)

4.7 The Heisenberg antiferromagnet

4.7.1 Introduction

The classical equivalent of the FM Weiss state for an antiferromagnet (AFM)is the Neel [11] state. With ordinary vectors substituting the spin operators,the configuration of minimum energy for (4.1) with J < 0 and B = 0 is clearlythat one in which all n.n. of a given spin are aligned exactly antiparallel to it,assuming that the lattice is bipartite. Let us consider this case, as we did inChap. 3, in which the A and B sub-lattices are anti-parallel at T = 0 in theclassical limit. Let us consider also a local anisotropy field Ba acting on eachspin, oriented along the local magnetzation, so that it reverts as we go froman A to a B site. This is the staggered field we considered in Chap. 3, with

4.7. THE HEISENBERG ANTIFERROMAGNET 93

the same magnitude on all sites. We add as well a uniform applied field. Theresulting Hamiltonian is:

H = J∑

<ij>

SiA· SjB

− γBa

∑

i

SziA −

∑

j

SzjB

− γB

∑

i

SziA

+∑

j

SzjB

(4.62)

We can easily extend the local boson HP transformation of the spin operators,applied before to a ferromagnet, to the case in hand. Let us choose the sublatticeA (B) with spin ↑ (↓). Then the natural choice for the boson operators is:

S+iA

=√

2S

√

1 − a†iai

2Sai

S+jB

=√

2Sb†j

√

1 −b†jbj

2S

SziA

= S − a†iai

SzjB

= −S + b†jbj

4.7.2 Antiferromagnetic spin-waves

As we did for the FM, we neglect all corrections to the linear terms in thetransformation equations above, and use Bloch’s theorem to Fourier transformthe local boson operators to running wave operators:

ck =1√N

∑

i∈A

eik·Riai

ai =1√N

∑

k

e−ik·Rick (4.63)

dk =1√N

∑

j∈B

eik·Rjbi

bj =1√N

∑

k

e−ik·Rjdk (4.64)

We remark that summations in Eqs. (4.63) and (4.64) are, as indicated, onlyon sublattice points. By definition, the sublattice has a larger lattice spacingthan the original lattice. In the special case of bipartite Bravais lattices, thevolume of the basic cell of each sublattice is twice that of the original atomic


lattice. Accordingly, since the k summations are performed on the first BZ of thesublattice, this zone has half the volume of the atomic BZ. Also, the number ofpoints N of each sublattice is half the number of magnetic atoms in the crystal.We assume that the lattice has inversion symmetry, so that the structure factoris symmetric:

γ(k) =1

ν

∑

δ

eik·δ = γ(−k) ≡ γk

where δ denotes a vector joining a site A (B) to a n.n. B (A) site on the othersublattice, and ν is the number of n.n. in the atomic lattice. We shall neglectfor the moment longer-range interactions. By following now the same procedureas for the ferromagnetic case, we obtain a quadratic Hamiltonian:

H(2)ex = −S2JνN − 2γBaSN + (JSν + γBa)

∑

k

(

c†kck + d†kdk

)

+γB∑

k

(

c†kck − d†kdk

)

+ JSν∑

k

γk

(

c†kd†k + ckdk

)

(4.65)


Let us simplify the notation. We define

ωe = JSν ; ωa = γBa ; ω0 = ωe + ωa

ω1(k) = γkωe ; ∆ = γB ; E0 = −S2JNν − ωaS

Then (4.65) is simplified as:

H(2)ex = E0 +

∑

k

ω0

(

c†kck + d†kdk

)

+ ∆(

c†kck − d†kdk

)

+∑

k

ω1(k)(

ckdk + c†kd†k

)

(4.66)

This is a quadratic form, but it is not yet diagonal, since the last term in (4.66)couples spin waves of both sublattices. In their original paper Holstein andPrimakoff [2] found a similar quadratic Hamiltonian upon considering simulta-neously ferromagnetic exchange and dipolar interactions. The dipolar terms, aswe shall see later on, generate non-diagonal terms very similar to those above.

In order to diagonalize such a quadratic form, HP introduced at this pointa linear canonical transformation of the original running wave operators, whichwas later re-discovered by Bogoliubov in connection with the theory of super-fluidity and superconductivity and is known as the Bogoliubov transformation[12]. Let us remark first of all that both c and d† have the effect of increasingSz

tot, while c† and d have the opposite effect. This suggests to define normalmodes created by a linear superposition of c and d†, since we are looking forcandidates to excited states of the original exchange Hamiltonian, which must


be also eigenstates of Sztot. Therefore, for each k let us perform the linear trans-

formation

αk = ukck − vkd†k

βk = Akdk +Bkc†k (4.67)

Let us for the time being suppress the index k, since we shall be workingin the subspace of the operators of a given k. We can choose the coefficientsu, v, A,B all real. The conditions we must impose upon the coefficients are:

1. the states created or annihilated by α and β must be linearly independent,which demands

[α , β ] = 0

This is satisfied if

A = u , B = −v (4.68)

2. the new operators must satisfy the canonical commutation relations

[

α , α†]

=[

β , β†]

= 1

which requires

u2 − v2 = 1 (4.69)

Exercise 4.15Prove (4.68) and (4.69).

Now we demand that, since α must annihilate a normal mode, it must be asolution of the Heisenberg equation of motion, with a real eigenfrequency:

ihdα(t)

dt= [Hex , α(t)] = λα(t) (4.70)

From (4.68), (4.69) and (4.70) and upon using the commutation relations for cand c† we arrive at a homogeneous system of linear equations for u and v:

(ω0 + ∆ − λ) u+ ω1(k)v = 0

ω1(k)u+ (ω0 − ∆ + λ) v = 0 (4.71)

with eigenvalues

λ = ∆ ±√

ω20 − ω2

1(k) (4.72)

The equations obtained for β = ud− vc† lead again to (4.72).


Exercise 4.16Verify (4.72) and the assertion above.

We now call the +(−) sign solution of (4.70) α(β). Introduce the notation

ω(k) =√

ω20 − ω2

1(k)

From (4.72) we get:(u

v

)

±=

−ω1(k)

ω0 ∓ ω(k)(4.73)

According to (4.69), we can introduce a parameter θ such that

u = cosh θ , v = sinh θ

Exercise 4.17Verify that only the solution with the + sign in (4.73) satisfies (4.69) and diag-onalizes the Hamiltonian (4.66).

With this choice, we have:

tanh 2θ = −ω1(k)

ω0(4.74)

Exercise 4.18By susbstituting the running wave operators ckand dkand their hermitian con-jugates in terms of αk and βk, obtain the diagonal Hamiltonian

H(2)ex = E0 − ω0N +

∑

k

(

α†kαk + 1/2

)

(ωk + ∆)

+∑

k

(

β†kβk + 1/2

)

(ωk − ∆) (4.75)

whereE0 = −SNωe − 2SNωa (4.76)

We have thus obtained two distinct AFM magnon branches, as linearly indepen-dent normal modes of the spin system in the harmonic approximation. One canaccordingly call this the free-spin-wave approximation (FSWA) for an AFM.For each particular k the corresponding term in the hamiltonian describes aquantized harmonic oscillator, with the 1/2 representing the contribution to theenergy of the zero-point quantum fluctuations. The new ground state energy,which is obtained when the occupation numbers for all k are zero, is the eigen-value of (4.75) for the true vacuum state, which we shall accordingly denote| 0 >. The expectation values of the number operators for this state vanish:

< 0 | α†kαk | 0 >=< 0 | β†

kβk | 0 >= 0 .


The renormalized ground state energy is accordingly

E′0 = E0 +

∑

k

(ωk − ω0) (4.77)


From (4.75) we see that the two branches have the respective eigenvalues

Eαk = ωk + ∆ , Eβ

k = ωk − ∆

and they are degenerate if the external field B is zero. If the anisotropy field isnon-zero, both branches have an energy gap ∆0 at zero external field:

∆0 =√

ωa(ωa + 2ωe) (4.78)

As B increases, there will be a critical B for which ∆ = ∆0 at which the systembecomes unstable because the lower branch energy vanishes. At this field thesystem goes into the spin flop phase, with all spins becoming approximatelyperpendicular to the applied field direction, which we already described inChap. 3 in the MFA. For Ba = B = 0, the two branches are degenerate.In particular for a one dimensional chain

ωk = ω0 | sin ka | (4.79)

where a is the distance from an A to a B site. For small k, and for a hypercubiclattice of any dimension, we have

ωk = vk + O(k3) (4.80)

where v is the magnon velocity.

4.7.3 Sublattice magnetization

The diagonalization process has led us to a redefinition of the operators whichcreate or annihilate elementary excitations, as we changed from ck, dk to αk, βk

(Bogoliubov transformation). We also verified that the ground state must havechanged in this process, to be consistent with the assumed properties of theoperators α and β. One consequence is that one finds deviations of the sublatticemagnetization from the saturation value S even at T = 0. To verify this let uscalculate the z component of the total up-sublattice spin:

SzA =

∑

i

SziA

= NS −∑

k

c†kck (4.81)

If we substitute in (4.81) c† and c in terms of the new operators, we need toinvert the Bogoliubov transformation.


Exercise 4.20Prove that the inverse Bogoliubov transformation is

ck = ukαk + vkβ†k

dk = vkαk + ukβ†k (4.82)

Denoting the new vacuum by | 0〉 we find that

< 0 | c†kck | 0 >= v2k < 0 | (1 + β†

kβk) | 0 >= v2k (4.83)

where we have explicitly imposed that βk annihilate | 0〉. We can easily calculatenow the sublattice deviation. We define the relative spin deviation for onesublattice as

∆M = 1− < SzA > /S

Then we have

∆M =∑

k

v2k = 1/2

∑

k

(

ω0√

ω20 − ω2

1(k)− 1

)

(4.84)


Let us define the anisotropy ratio

α =ωa

ωe(4.85)

Exercise 4.22Calculate ∆M for a one dimensional lattice, and show that for small anisotropyratio α

∆M = − 1

2S

(

1 +ln 2α

π

)

(4.86)

Eq. (4.86) shows that in the absence of anisotropy the FSWA excludes longrange AFM order in one dimension, even at zero T : one is led to conclude that,since the spin deviation at T = 0 diverges, the ordered Neel ground state is notthe correct one. If the LRO parameter σ = 0 we must re-formulate the wholetheory, since the notion of a spin-wave is based upon a translationally invariantNeel ground state. This we already mentioned in our reference to the work ofBethe [3], who proved that the ground state of the AF chain has no LRO.Numerical calculations yield for S∆M , within the FSWA, the values:

• 0.197 for the square lattice;

• 0.0078 for the NaCl magnetic lattice;


• 0.0593 for the body-centered atomic lattice.

Let us remark that in a lattice with a magnetic NaCl structure, the first n.n. ofeach spin form a simple cubic coordination, so that this structure is sometimescalled simple cubic for short, in the present context.We notice that the departure of the sublattice magnetization from the Neelsaturation value at T = 0 is much greater in low dimensions than in 3d.

4.7.4 Ground state energy of AFM

It is interesting to compare (4.76) with the corresponding energy of the classicalNeel state. As we did for the FM, in the semi-classical limit we substitute eachspin operator by an ordinary vector of length

S0 =√

S(S + 1)

Then the classical energy in the Neel configuration without external field is

Eclass0 = −JNνS(S + 1) − 2γBa

√

S(S + 1)N (4.87)

With the help of (4.76) and (4.87) we calculate the difference between the groundstate energy in the harmonic approximation and the semi-classic energy of theNeel state in the S → ∞ limit:

EFSWA0 −Eclass

0 =∑

k

ωk + O(1/S) (4.88)

This result is exactly what should be expected: the quantization of the spinfluctuations in the harmonic approximation yields a ground state energy whichis higher than the classical counterpart by a magnitude equal to the energy ofthe zero-point motion of all the harmonic modes, just as in the ferromagneticcase.

All these estimates are based on the independendent harmonic oscillator ap-proximation or FSWA. However, more general conclusions have been arrived atregarding the ground state of the Heisenberg Hamiltonian. Under not too re-strictive conditions, Lieb and Mattis [17] proved that it must be a singlet. Thenit was also proved [18] that in such a case time reversal symmetry demands thatthe expectation value in the ground state < Sz

n >= 0 for all sites n, at oddswith Neel’s state.

In spite of this, real antiferromagnets are Neel like. One can argue, fol-lowing Nagamiya et al. [14], that there is scarcely any coupling between bothdegenerate Neel states coupled by time reversal symmetry. An estimation byAnderson [19] of the switching time from one orientation to the time reversedone yields 3 years, in the absence of anisotropy. The non-zero anisotropy presentin practically all systems would tend to make the total spin reversal even moredifficult, improving the metastability of the sublattice pattern. Some variationalcalculations of the energy of the ground state, which assumes it to be a singlet,give similar results to those performed within the two sublattice model [20],which can be taken as an indication that the resonance energy between the twodegenerate Neel states is indeed very small.


References

1. Mermin, N. D. and Wagner, H. (1966) Phys. Rev. Lett. 17, 1133.

2. Holstein, T. and Primakoff, H. (1940) Phys. Rev. 58, 1098.

3. Bethe, H. A. (1931) Z. Physik 71, 205.

4. Mattis, D. (1963) Phys. Rev. 130, 76.

5. J.C.Slater, J. C.(1930) Phys. Rev. 35, 509.

6. Bloch, F. (1930) Z. Physik 61, 206.

7. Moller, C. (1933) Z. Physik 82, 559.

8. Keffer, F. (1966) Handb. der Phys. 2 Aufl., Bd.XVIII, Springer–Verlag.

9. Klein, M. J. and Smith, R. S. (1950) Phys. Rev. 80, 111.

10. Keffer, F., Kaplan, H. and Yafet, Y. (1953) Am. J. Phys. 21, 250.

11. Neel, L. (1932) Ann. Phys. 18, 5; (1948) 3, 137; (1936) Compt. rend.203, 304.

12. Bogoliubov, N. N. (1947) J. Phys. USSR 9, 23; (1958) Nuovo Cimento 7,794.

13. Herring, C. and Kittel, C. (1951) Phys. Rev. 81, 869; (1952) id. 88,1435.

14. Nagamiya, T., Yosida, K. and Kubo, R. (1955) “Antiferromagnetism”,Adv. in Phys. 4, 1-112.

15. Kubo, R. (1952) Phys. Rev. 87, 568.


17. Lieb, E. and Mattis, D. (1962) Phys. Rev. 125, 164;(1962) J. Math.Phys. 3, 749.

18. Pratt jr., G. W. (1961) Phys. Rev. 122, 489.


20. Marshall, W. (1955) Proc. Roy. Soc. (London) A 232, 48.

Chapter 5

Magnetic anisotropy

5.1 Introduction

We have already mentioned anisotropy effects in several previous sections. Letus now study the microscopic origin of magneto-crystalline anisotropy terms inferromagnets. First, let us determine the general structure of the anisotropicterms in the free energy. We assume for simplicity that the magnetization Mis uniform inside the sample, thereby discarding the presence of domains. Atgiven T and B we can expand Helmholtz free energy F in powers of M[1]:

F = F0 + Fa (5.1)

where F0 is isotropic and the rest of the terms involve anisotropic polynomialsof increasing degree in the components of M. The free energy must be invariantunder time- reversal (M → −M) so only even powers of M can appear in theexpansion.

It is sometimes convenient to change variables to

ui = Mi/ | M |≡ αi (5.2)

where αi are the director cosines of M referred to cartesian crystalline axes.Here, of course

∑3i=1 α

2i = 1.

Both F0, and Fa must be invariant under all the operations of the symmetrygroup of the crystal.

As an example, in the case of uniaxial rotational symmetry it is convenientto express M in the spherical components Mz,M± = Mx ± iMy, so that if thesymmetry group contains rotations of the form 2πm/n , where (n,m) = integersand m = 1, · · · , n − 1 , we can include in the expansion terms like Re(M p·n

± ).In this case it is more convenient to express the components of the unit vector uin terms of the spherical angles θ (latitude) and φ ( azimuthal angle, measuredcounterclockwise on the z = 0 plane from the semiaxis of positive x ). θ = 0is the positive z semiaxis, while φ = 0 is the positive x semiaxis. A real even

101

102 CHAPTER 5. MAGNETIC ANISOTROPY

invariant function has the form:

Γ2p(θ, φ) = 1/2[

(M+)2p + (M−)2p]

= sin2p θ cos 2pφ

or z2p or a product of both forms.Let us review the forms of the expansion of F for different symmetry groups.

We shall use Schoenflies point group notation in the following[2].

Uniaxial tetragonal symmetry : C4h group.Here n = 4. We assume there is a mirror symmetry plane perpendicular to

the rotation symmetry axis, and this is symbolized by the subindex h in C4h.The expansion of F is

F = F0 +A2 sin2 θ +A4 sin4 θ +B4 sin4 θ cos4 φ+ · · · (5.3)

where the dots contain higher order polynomials.

Uniaxial hexagonal symmetry: C6h group.For n = 6 every term in the free energy expansion must be invariant under

rotations around the z axis by 2πm/6, with m = 1, 2, · · ·5, mirror reflectionin the (x, y) plane and inversion at the origin (equivalent to time reversal).Therefore we have

F = F0 +A2 sin2 θ +A4 sin4 θ +A6 sin6 θ cos 6φ+ · · · (5.4)

Upon expressing sin2n θ in terms of cosmθ , m = 2, 4, · · · , 2n we find :

F = F0 +1

2Ku1(1 − cos 2θ +

1

8Ku2(3 − 4 cos 2θ + cos 4θ)

+1

32Ku3(10 − 15 cos 2θ + 6 cos4θ − cos 6θ)

+1

32Ku4(10 − 15 cos 2θ + 6 cos4θ − cos 6θ) cos 6φ+ · · · (5.5)

The u subindex above refers to uniaxial symmetry.It is customary to quote anisotropy energy per unit volume. The coefficients

in Eq. (5.5) and similar expansions contain powers of |M|, so they depend ontemperature. Numerical values for Co at 288K are[3]:

Ku1 = 4.53× 106 erg cm−3

Ku2 = 1.44× 106 erg cm−3

Higher order terms are smaller. Since these coefficients are positive the axis ofeasy magnetization for Co is the hexagonal c axis. When they are negative thez (or c) axis is a hard axis. In the latter case the free energy is minimized whenθ = π/2 and the magnetization lies in the c plane (orthogonal to the c axis),which is therefore an easy plane. If Ku1 and Ku2 have different signs the stabledirection of M lies on an easy cone.

5.1. INTRODUCTION 103

Cubic symmetry: Oh groupThe symmetry operations were described in Sect. 1.4.1. The first anisotropic

term comes in fourth order. We find, up to eighth order :

F = F0 +K1(α21α

22 + α2

2α23 + α2

3α21) +K2(α

21α

22α

23)

+ K3(α21α

22 + α2

2α23 + α2

3α21)

2 + · · · (5.6)

where Ki are called cubic anisotropic constants [3]. For Fe at 293K ,

K1 = 4.72× 105 erg cm−3

K2 = −0.075× 105 erg cm−3

while for Ni at 296K ,

K1 = −5.7× 104 erg cm−3

K2 = −2.3× 104 erg cm−3

Along the [100] and equivalent directions, the anisotropy contribution vanishesindependently of the values of the coefficients. For the [111] directions we have

Fa =1

3K1 +

1

27K2 +

1

9K3 + · · · (5.7)

If K1 > 0 as in Fe, [100] is the easy axis, if we neglect higher order terms. InNi, where K1 < 0 the set of the [121] directions are easy. Let us consider themagnetization on the (001) plane (the y − z plane). If θ=angle between themagnetization and the positive x axis, we have α1 = cos θ , α2 = sin θ , α3 = 0.Then Eq. (5.6) becomes:

Fa =1

8K1(1 − cos 4θ) +

1

128K3(3 − 4 cos 4θ + cos 8θ) + · · · (5.8)

with no contribution from K2


If M lies on the (110) plane, and we call θ the angle it makes with the pos-itive z axis, we obtain

Fa =K1

32(7 − 4 cos 2θ − 3 cos 4θ)

+K2

128(2 − 4 cos 2θ − 2 cos 4θ)

+1

2048K3 (123− 88 cos 2θ − 68 cos 4θ + 24 cos 6θ + 9 cos 8θ)

+ · · · (5.9)



Let us now calculate the anisotropy energy with M on the (111) plane. Onecan use a new reference frame with z′ along the [111] axis, x′ parallel to the[110] axis and y′ in the (110) plane. Let θ be the angle between M and x′ inthe x′ − y′ plane. Then, α′

1 = cos θ, α′2 = sin θ, α′

3 = 0. We obtain :

Fa =K1

4+K2

108( 1 − cos 6θ ) +

K3

16+ · · · (5.10)


The results obtained show that in all cases considered so far Fa can be writtenas

Fa =∑

n

A2n(φ) cos 2nθ (5.11)

For example, for uniaxial symmetry we obtain, upon comparing Eq. (5.11) withEq. (5.5):

A2 = − (Ku1 +Ku2)

2− 15

32Ku3( 1 + cos 6φ )

Exercise 5.4Obtain the form of the A2n in terms of the coefficients defined previously, forthe different cases considered above.

We have up to this point described the general form of the anisotropic termsin the series expansion of Helmholtz free energy for several symmetry types offerromagnets. It remains now to derive the quantum mechanical origin of theseterms.

5.2 Microscopic origin of anisotropy

We obtained in Sect. 1.7 the form of the single-ion anisotropy energy operatoras one of the effects of the spin-orbit interaction. Therein we considered per-turbation terms up to second order. We must go to forth order to find relevantsingle-ion anisotropy contributions to the single ion anisotropy energy if the ionlattice is cubic, since in this case the second order correction to the gyromag-netic ratio g isotropic. The fourth order single-ion terms are usually written inthe form:[4]

Heff =a

6S4

x + S4y + S4

z − 1

5S(S + 1)[3S(S(S + 1) − 1] (5.12)

5.2. MICROSCOPIC ORIGIN OF ANISOTROPY 105

These terms are only relevant if S > 1, since otherwise the operator above re-duces to a constant.

It turns out that the spin-orbit force also generates the anisotropic inter-actions among spins which are at the origin of the anisotropy energy that wedescribed phenomenologically in the previous section.

Let us consider a pair of magnetic ions, which can be in general of differentatomic species, each of which has a non-degenerate ground state Γi

0 , i = 1, 2.The relevant interaction terms are

V = λ(~L1 · ~S1 + ~L2 · S2) + Vex (5.13)

We need to go now up to third order in V to obtain an effective spin-spinanisotropic Hamiltonian. Consider the following processes :

• the ~L1 · ~S1 term excites ion 1 from Γ(1)0 to an excited state n1.

• the exchange interaction couples the excited ion 1 to ion 2 in its ground

state Γ(2)0 .

• the ~L1 · ~S1 term returns ion 1 to its ground state.

These processes are contained in the third order perturbation expansion of thecomplete Hamiltonian for the pair of ions.

The stationary Schrodinger equation for the two spin system reads :

H | ψ〉 = (H0 + εV ) | ψ〉 = E | ψ〉 (5.14)

where the auxiliary parameter ε is set equal to 1 at the end of the calculation.Just as in Sect. 1.7 we perform a standard perturbation expansion of H inpowers of ε in the orbital subspace of the ions, whereby spins are treated asc-numbers. We simplify the notation and express the formal expansion as :

E = E0 +

∞∑

n=1

εnE(n)

| ψ〉 = | 0102〉 +

∞∑

n=1

εna(1)(n) | n1 02〉 + a

(2)(n) | 01 n2〉 (5.15)

where indices 0i, ni refer to ground and excited states of ion i and superindicesin the coefficients denote the different ions. We disregard terms where both ionsare excited, since these should be smaller.Contributions to the expansion from both ions are additive, and we also have

〈n102 | 0102〉 = 〈01n2 | 0102〉 = 0

All states are assumed normalized to 1. The same normalization condition isimposed on | ψ〉.The third order correction from ion (1) is

∆(1)3 =

∑

n1n2

〈0102 | V | n102〉〈n102 | V | n′102〉〈n′

102 | V 0102〉(E

(1)0 −En1)(E

(1)0 −En′

1)

(5.16)


where we allow for the ions being different and having consequently a differentground state energy.

Exercise 5.5Obtain Eq. (5.16) above.

Since we are looking for a generalization of exchange interactions we selectin particular terms which are of first order in Vex, with matrix elements of theform

〈n102 | Vex | n′102〉 = −J (n102;n

′102)S1 · S2 (5.17)

where the sign is chosen so as to make the interaction ferromagnetic when J > 0.Apart from factors which depend on the overlap integral between the ions (seeSect. 2.2) the matrix element in Eq. (5.17) is proportional to

∫

d3r1 d3r2 φ

∗n1

(r1 − R1)φ∗0(r2 −R2)

1

r12φ0(r1 − R2)φn′

1(r2 −R1) (5.18)

After some algebraic manipulations, the third order effective spin exchange in-teraction can be expressed as ( with µ, ν = x, y, z) :

∆E3 = −∑

µν

[

Sµ1 Γ(1)

µν S1 · S2Sν1 + Sµ

2 Γ(2)µν S1 · S2S

ν2

]

(5.19)

where we have added together terms from both ions and

Γ(1)µν = λ2

∑

n1,n′1

(0 | Lµ | n1)J (n102;n′102)(n

′1 | Lν | 0)

(

En1 −E(1)0

)(

En′1−E

(1)0

) (5.20)

with the corresponding expresion Γ(2)µν for ion 2.

Exercise 5.6Obtain Eq. (5.20).

In the case S1 = S2 = 1/2 the expression (5.19) reduces to

Haniso = −1

4

∑

µν

∑

i=1,2[

Γ(i)µν + Γ(i)

νµ − δµν

(

Γ(i)xx + Γ(i)

yy + Γ(i)zz

)]

Sµ1 S

ν2 (5.21)

This is called pseudo-dipolar interaction because it has the same form of thedipolar one (see Chap. 7).


5.2. MICROSCOPIC ORIGIN OF ANISOTROPY 107

This interaction is symmetric under the interchange of spins 1 ↔ 2, but onecan select some perturbation terms of second order in V , which are bilinearin the spin-orbit and the exchange potentials, and generate the pseudo-scalar,Dzialoshinsky-Moriya (DM) [5, 6] anti-symmetric interaction, as shown below.Let us consider the second order contribution

∆E2 = −λ∑

n1

1

En1 −E(1)0

· ( 〈0102 | Vex | n102〉〈n1 | L1 | 01〉 · S1

+ 〈01 | L1〉 · S1〈n1 | Vex | 0102〉 ) + · · · (5.22)

where dots stand for a term of the same form involving excited states of ion 2.Since L is pure imaginary and hermitean, we have:

〈01 | L1 | n1〉 = 〈n1 | L1 | 01〉∗ = −〈n1 | L1 | 01〉 .

Assume the matrix elements of the exchange interaction are real. Then

∆E2 = −λ∑

n1

J (n102 ; 0102)〈01 | Lµ1 | n1〉 [ Sµ

1 , (S1 · S2) ](

En1 −E(1)0

) + · · · (5.23)

We use now the identity

[ S1, (S1 · S2) ] = −iS1 ∧ S2

and obtain the DM interaction as:

HDM = λD · S1 ∧ S2 (5.24)

where

D = −iλ

∑

n1

J (n102 ; 0102)〈01 | L1 | n1〉(

En1 −E(1)0

)

−∑

n2

J (01n2 ; 0102)〈02 | L2 | n2〉(

En2 −E(2)0

)

(5.25)

If the middle point of ions 1 and 2 is a center of symmetry, interchanging theions is a symmetry operation, and D = 0. Otherwise, in general D 6= 0.Rules concerning the direction of D for different symmetries were derived byMoriya [6].

Let us return to the general form of the anisotropic exchange interaction,Eq. (5.19). The r. h. s. has the form of a fourth order polynomial in thecomponents of neighbouring spins, of the form :

Sµ1 ΓµνS

α1 S

α2 S

ν1


In a uniformly magnetized ferromagnet, and within the mean field approxima-tion, we can substitute the spin components by their thermodynamic averages,which are proportional to the corresponding components of the magnetization,and we obtain for the anisotropy energy the form

Eanis = ΓµνMµ Mν MαMα (5.26)

where repeated indices are summed overi.For cubic symmetry

Γµν = Γ · δµν

so that one gets, appart from trivial isotropic terms:

Eanis = Γ[M2xM

2y +M2

yM2z +M2

zM2x +M4

x +M4y +M4

z ] (5.27)

Allowing for other sources of fourth order contributions, the general expressionfor cubic symmetry is:

Eanis = A[M2xM

2y +M2

yM2z +M2

zM2x ] + B[M4

x +M4y +M4

z ] (5.28)

The second term above can be reduced to the same form as the first by usingthe identity, satisfied by the director cosines:

α41 + α4

2 + α43 = 1 − 2(α2

1α22 + α2

2α23 + α2

3α21)

so that we obtain the form of the fourth order term in eq. (5.6).

5.3 Magneto-elastic coupling

Inside one particular domain in a FM the magnetization is uniform, except ofcourse at the boundaries. We assume orthogonal crystal axes.

Let us consider a given pair of spins located on opposite ends of the bondr = n1 x +n2 y + n3 z in terms of the unit vectors of the three orthogonal axes.Inside a uniformly magnetized domain, spins are parallel. Their interactionenergy, including up to pseudo- quadrupolar terms, can be written as:

w(r) = g(r) + l(r)(cos2(φ) − 1/3)

+q(r)(cos4(φ) − 6/7 cos2(φ) + 3/35 ) (5.29)

where φ is the angle between the magnetization direction and the bond r, and theexpansion is in terms of Legendre polynomials. The range functions g(r), l(r)and q(r) correspond to the isotropic effective exchange, dipolar and quadrupolarinteractions respectively. In the following we neglect quadrupolar terms.

The isotropic term g(r) in a strained crystal may only change if the lengthof the crystal bonds change, while the dipolar and quadrupolar terms will alsochange upon rotational deformation of the lattice. We shall see below that theeffect of strain on the dipolar terms can induce changes in the orientation of themagnetization.

5.3. MAGNETO-ELASTIC COUPLING 109

Under strain, the unit vectors of the crystal axes are transformed into theslightly deformed vectors [7]:

x′ = (1 + εxx)x + εxyy + εxzz

y′ = εyxx + (1 + εyy)y + εyzz

z′ = εzxx + εzyy + (1 + εzz)z

where εαβ are components of the symmetric second rank strain tensor. Weconstruct now the tensor obtained by calculating all the scalar products amongthe strained unit vectors (the strain tensor components, are assumed small, sowe neglect second order terms):

exx ≡ (x′)2 ≈ 1 + 2εxx

exy ≡ x′ · y′ ≈ εxy + εyx = 2εxy

and the corresponding expressions for the other components, which are obtainedby permutation. The elastic energy of the crystal can be expressed in terms ofthis tensor as [8]

Eel =1

2c11∑

α

e2αα +1

2c44∑

α6=β

e2αβ

+ c12∑

α6=β

eααeββ (5.30)

We shall now calculate the magneto-elastic energy. We consider separately thecontributions of pairs of spins connected by bonds along each crystal axis. Ifthe bond is along the x axis, the director cosines of the strained unit vector x′

referred to the unstrained crystal axes, again to first order in the strain tensor,are:

β1 ≡ x′ · x = 1 + εxx

β2 ≡ x′ · y = εxy

β3 ≡ x′ · z = εxz (5.31)

The interaction energy is obtained as the change of the magnetic energy due tothe changes in the length and the orientation of the bond vector under strain.If the bond length along x is a0 before straining, under strain the new bondlength is

a = a0

(

1 +1

2exy

)

We can write the effective dipolar interaction energy for the pair of first nearestneighbour spins along the x axis as :

wx = l(a)

(

∑

i

αiβi)2 − 1/3

)

(5.32)


where a and βi under strain were given above and αi are the director cosinesof the magnetization. We can obtain the change in the magnetic interactionenergy for this particular pair as

δwx = a0

(

∂l

∂a

)

ao

exx(α21 − 1/3) + l(a0)α1α2 exy + l(a0)α1α3 exz (5.33)

For a simple cubic magnetic lattice the total change of dipolar energy per unitvolume is the sum of the three terms obtained by permuting the x axis inEq. (5.33) with y or z, and the result is

Emag−el = B1

∑

ν=x,y,z

eνν(α2ν − 1/3) +B2

∑

ν 6=µ

eνµαναµ (5.34)

with

B1 = Na0

(

∂l

∂a

)

ao

B2 = 2N l(a0) (5.35)

where N = the number of atoms per unit volume.

Exercise 5.8Prove Eqs. (5.35).

The same expression (5.34) is valid for the other two cubic lattices, if the coef-ficients are defined as

B1 =N

2

(

6l(a0) + a0

(

∂l

∂a

)

ao

)

B2 = N

(

2l(a0) + a0

(

∂l

∂aao

))

(5.36)

for the f.c.c. lattice, and

B1 =8N

3l(a0) , B2 =

8N

9

(

1 + a0

(

∂l

∂a

)

ao

)

(5.37)

for the b.c.c. lattice.

Exercise 5.9Prove Eqs. (5.36), (5.37).

We must add to the magneto-elastic energy in Eq. (5.34) the purely elasticenergy from Eq. (5.30) in order to obtain the total energy:

Etot = Eel +Emag−el (5.38)

5.4. MAGNETOSTRICTION 111

The strain tensor components in equilibrium are obtained by imposing that thetotal energy be stationary:

∂Etot

∂eαα=∂Etot

∂eαβ= 0 , α , β = x, y, z (5.39)

Exercise 5.10Show that the solution to Eqs. (5.39) is:

exy = −B2α1α2

c44

eyz = −B2α3α2

c44

ezx = −B2α1α3

c44

eνν =B1

c12 − c11

(

α2ν − 1/3

)

(5.40)

Remark that the solution above implies∑

ν eνν ≡ dilation = δV/V = 0 [7].

5.4 Magnetostriction

In the presence of magnetization the length l of an element of material expe-riences a relative change of the order of δl/l ∼ 10−5 − 10−6, which althoughsmall has an important influence on the domain structure and on the “techni-cal magnetization”, which is the process whereupon the magnetization M(H)grows under the application of an external field H until it reaches saturation.[9]Let us consider a uniformly magnetized sample. In this case we also expect auniform elastic deformation, as described in the previous section. The relativeelongation of a segment of length l oriented in the direction of the unit vectorn = (β1, β2, β3) is the same as the elongation of n, namely:

δl

l= n′ · n− 1 =

∑

i

β2i eii +

∑

i6=j

βiβjeij (5.41)

which reads, upon substitution of the equilibrium values for the strain tensorcomponents given in Eqs. (5.40):

δl

l=

B1

C12 − C11

(

∑

i

α2i β

2i − 1/3

)

− B2

C44

∑

i6=j

αiαjβiβj (5.42)


If the domain magnetization direction is [100] the elongation in the same direc-tion is :

(

δl

l

)

100

≡ λ100 = 2/3B1

C12 − C11(5.43)

while if both are in the [111] direction we get

(

δl

l

)

111

≡ λ111 = −1/3B2

C44(5.44)

If λ100 = λ111 the magnetostriction is isotropic. For this case, the uniformrelative elongation (5.42) for a general direction becomes :

δl

l=

3

2λ(cos2 θ − 1/3) (5.45)

where θ = angle between the strain which is being measured and the magneti-zation direction.

In the general case, we can substitute in Eq. (5.42) the coefficients B1 andB2 in terms of λ100 and λ111:

δl

l=

3

2λ100

(

∑

i

α2i β

2i − 1/3

)

+ 3λ111

∑

i6=j

αiαjβiβj (5.46)

For a polycrystalline sample, we must perform the average of Eq. (5.46) forαi = βi over the unit sphere. The result is

λ =2

5λ100 +

3

5λ111 (5.47)


For the cubic lattices, the fractional elongation along any direction can be ex-pressed in terms of the ones along those two particular ones. For instance, weget

λ110 =1

4λ100 +

3

4λ111 (5.48)

The interested reader can find in Ref. [9] detailed information and many refer-ences on the magnetostrictive properties of specific magnetic materials, as wellas on several techniques for measuring λ100 and λ111.

5.5. INVERSE MAGNETOSTRICTION 113

5.5 Inverse magnetostriction

Just as the presence of magnetization induces deformation of the crystal lat-tice, the inverse also occurs, namely, changes in the magnetization caused by anexternal stress. We have derived the expression (5.34) for the magneto-elasticenergy, which couples magnetization to strains. We could obtain from it thecoupling of the magnetization with the stress components, if we make recourseto the equations of elasticity that relate both tensors.We call the stress components Xx, Xy, ..., Yx · · ·, where Xα is the force per unitsurface oriented perpendicular to axis x in the direction α = x, y or z, etc. [7]We defined the strain tensor components in Eq.(5.30) as a scalar product, whichis symmetric. Therefore it has only 6 independent components. We can take ad-vantage of this to simplify the notation. Let us make the following substitutionsfor the indices:

xx ≡ 1, yy ≡ 2, zz ≡ 3

yz ≡ 4, zx ≡ 5, xy ≡ 6

The linear relation between stress and strain is

e = SX (5.49)

where the elements of S are called the elastic compliance constants, or in shortjust the elastic constants. The inverse relation is written as

X = Ce (5.50)

and the elements of C are the elastic stiffness constants, or moduli of elasticity.Clearly,

S = C−1 (5.51)

The elastic energy can be written as a quadratic form in the six independentcomponents of the strain tensor as

Eel =1

2

∑

λ,µ

Cλµeλeµ (5.52)

Stresses are canonically conjugate to strains. For instance,

Xx =∂Eel

∂e1= C11e1 +

1

2

6∑

µ=2

(

C1µ + Cµ 1

)

eµ (5.53)

Comparison with (5.50) yields:

Cλµ =1

2

(

Cλµ + Cµλ

)

= Cµλ (5.54)

This symmetry reduces the number of independent elements of the 6 × 6 Cmatrix from 36 to 21 in the most general case. For cubic symmetry there


are many more constraints, and one is left with only 3 independent stiffnessconstants, the same being true for the inverse matrix S of the elastic constants.The structure of the C tensor for a cubic lattice is:

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12 C11 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C44

(5.55)

The inverse S of this matrix, namely the elastic compliance tensor for the cubiclattice is:

S11 S12 S12 0 0 0S12 S11 S12 0 0 0S12 S12 S11 0 0 00 0 0 S44 0 00 0 0 0 S44 00 0 0 0 0 S44

(5.56)

Exercise 5.12Prove the following relations:

S44 =1

C44; S11 − S12 =

1

C11 − C12

Let us suppose that a uniaxial stress of magnitude σ (in units of pressure)is applied on the ferromagnet in the direction with cosines (γ1, γ2, γ3). Thestress tensor components are therefore:

Xij = σ γiγj = Xji (5.57)

The strains can be calculated in terms of the applied stresses:

exx = σ(

S11γ21 + S12( γ

22 + γ2

3 ))

exy = σS44γ1γ2 (5.58)

We substitute now these expressions for the strains into Eq. (5.34) and we obtainthe magnetoelastic energy under a uniaxial stress as:

Emag−el = σB1(S11 − S12)

(

3∑

i=1

γ2i α

2i − 1/3

)

+σB2S44

3∑

i6=j

γiαiγjαj (5.59)

5.6. INDUCED MAGNETO-CRYSTALLINE ANISOTROPY 115

Upon use of Eqs. (5.43, 5.44) to eliminate B1 and B2 we arrive at:

Emag−el = −3

2σλ100

(

3∑

i=1

γ2i α

2i − 1/3

)

−3σλ111

3∑

i6=j

γiαiγjαj (5.60)

In Fe, K1 > 0, and the easy axis is along one of the cubic axis. For M alignedin the [100] direction, we have α1 = 1, α2 = α3 = 0, and

E[100]) = −3

2σλ100(γ

21 − 1/3) (5.61)

Exercise 5.13Show that for M in the [111] direction we get

E[111](σ) = −3

2σ(cos2(φ) − 1/3) (5.62)

where φ = angle between the uniaxial stress and [111].

We see that in the isotropic case the same form of Eqs. (5.61, 5.62) applieswith λ = λ100 = λ100.

In conclusion, the isotropic magnetostriction energy generates in this casea uniaxial anisotropy term in the spin Hamiltonian like the one we consideredseveral times before, with the easy axis in the direction of the applied stress.

5.6 Induced magneto-crystalline anisotropy

We have written in 5.40 the explicit expressions for the equilibrium componentsof the strain tensor in a uniformly magnetized material, in terms of the directorcosines of M referred to the cubic crystal axes. If one substitutes these expres-sions for the strain components into (5.30) and (5.34) for the two parts of thetotal energy, after some straightforward algebra one finds the result:

Etot(αi) = ∆K1(α21α

22 + α2

2α23 + α2

3α21 ) (5.63)

where we have defined

∆K1 ≡ −(

B21

C12 − C11+

B22

2C44

)

(5.64)

The conclusion is that the crystal deformation produced by the magnetizationinduces, via the magneto-elastic coupling, a magneto-crystalline anisotropy en-ergy of the same form as the one found in Sect. 5.2. Since both effects acttogether, the measured values of K1 must be corrected by subtracting ∆K1 ifwe want to obtain quantitative information on the spin-orbit coupling constantΓ defined therein [9].


References

1. Van Vleck, J. H. (1937) Phys. Rev. 52, 1178.

2. Landau, L. D. and Lifshitz, E. (1966) “Mecanique Quantique”, MIR Edi-tions, Moscow.

3. Chikazumi, S. (1997) “The Physics of Ferromagnetism”, second edition,Chap. 12, Clarendon Press, Oxford.

4. Yosida, Kei (1998) “Theory of Magnetism”, Chap. 3, Springer.

5. Dzyaloshinsky, I. (1958) J. Phys. Chem. Solids 4, 241.

6. Moriya, T. (1960) Phys. Rev. 120, 91.

7. Kittel, Charles (1971) “Introduction to Solid State Physics”, fourth edi-tion, Chap. 4, John Wiley & Sons, Inc.

8. C. Kittel, loc. cit. pg. 140.

9. Chikazumi, S., loc. cit., Chap. 14.

Chapter 6

Green’s Functions Methods

6.1 Definitions

Many physical properties of magnetic systems depend on, or can be expresedin terms of, the correlation between pairs of spin operators. In turn, thesequantities can be obtained from the corresponding Green’s functions. Let usfirst remind the definitions of both. In the Heisenberg picture, the time depen-dent operators which correspond to physical observables are obtained from thecorresponding constant (in time) Schrodinger picture operators as

A(t) = eiHt/hAe−iHt/h (6.1)

which implies

ihdA(t)

dt= [ A(t), H ] (6.2)

We define the correlation function of two operators A and B as:

FAB(t, t′) ≡ 〈 A(t)B(t′) 〉 (6.3)

where

〈 X 〉 =Tr(e−βHX)

Tr(e−βH)(6.4)

Exercise 6.1Prove that, if H has no explicit time dependence,

FAB(t, t′) = fAB(t− t′) (6.5)

We can define other functions of t and t’, satisfying equation (6.2) with a sin-gular inhomogeneous right hand side term, which would correspond to Green’sdifferential equation associated with (6.2). To this end, let us first remark that

ihd〈 A(t)B(t′) 〉

dt− 〈 [ A(t), H ]B(t′) 〉 = 0 (6.6)

117

118 CHAPTER 6. GREEN’S FUNCTIONS METHODS

We can now introduce some new quantities related to the ones above, which aredenoted by 〈〈 A;B 〉〉, where:

〈〈 A;B 〉〉 ≡ −iθ(t− t′)〈 [A(t), B(t′)] 〉 (6.7)

These are functions of t and t′ which satisfy:

ihd〈〈 A;B 〉〉

dt− 〈〈 [ A(t), H ];B 〉〉 = hδ(t− t′)〈〈 A;B 〉〉 (6.8)

It is convenient to choose A and B such that, besides their being directlyconected with the observables one wants to calculate, one has a simple enoughinhomogeneous term in (6.8). One very convenient choice would be any oneleading to a constant factor multiplying Dirac’s δ in (6.8). If A and B areone-particle boson (or fermion) operators, their conmutator (or anticonmutatorrespectively) has this property. So, we are led to:

GrAB(t, t′) = −iθ(t− t′)〈 [ A(t), B(t′) ]η 〉 (6.9)

where

θ(x) =

1 x ≤ 00 x < 0

, η =

−1 fermions+1 bosons

(6.10)

and [ A,B ]η ≡ AB − ηBA. We also define the advanced Green’s function as

GaAB(t, t′) = iθ(t′ − t)〈 [ A(t), B(t′) ]η 〉 (6.11)

G(r,a)AB are bi-linear funtionals of the operators A and B.


ih∂G

(r,a)AB (t, t′)

∂t= hδ(t− t′)〈 [ A(t), B(t′) ]η 〉

+ 〈〈 [ A(t), H ];B(t′) 〉〉(r,a) (6.12)

Indices “r” and “a” refer to retarded and advanced respectively. The causalGreen’s function is defined as:

G(c)AB = −i〈 Tη(A(t)B(t′)) 〉 (6.13)

where

Tη (A(t)B(t′)) ≡ θ(t− t′)A(t)B(t′) + ηθ(t′ − t)B(t′)A(t) (6.14)

Tη is called the “time ordering” operator.

6.2. SPECTRAL REPRESENTATION 119


ih∂G

(j)AB(t, t′)

∂t=

hδ(t− t′)〈 [A,B]η 〉 + 〈〈 [A(t), H ];B(t′) 〉〉(j) (6.15)

with (j) = (r, a, c).

We define the Fourier transform:

G(ω) =1

2π

∫ ∞

−∞

eiωtG(t)dt (6.16)

with

G(t) =

∫ ∞

−∞

e−iωtG(ω)dω (6.17)

So that the Fourier transform of (6.12) is:

ωG(j)AB =

1

2π〈 [A,B]η 〉 + 〈〈 [A,H ];B 〉〉ω (6.18)

6.2 Spectral representation

Let us consider the eigenvalue equation of an arbitrary Hamiltonian:

H | n 〉 = En | n 〉 (6.19)

where n represents the set of all quantum numbers which define the eigenkets.In general, the evaluation of the previously defined Green’s functions involvesthe calculation of averages of products of certain operators A and B evaluatedin the Heisenberg picture, that is of quantities like:

〈 B(t′)A(t) 〉 ≡ Z−1∑

n

e−βEn < n | ei Hh

t′Be−i Hh

t′ei Hh

tAe−i Hh

t | n >=

Z−1∑

n

e−βEnei E−nh

t′ < n | B | m > ×

e−i Emh

(t−t′) < m | A | n > e−i Enh

t =

Z−1∑

n,m

ei (En−Em)h

(t′−t)e−βEn ×

< n | B | m >< m | A | n > (6.20)

We call τ ≡ t− t′ and then we Fourier transform (6.20) with respect to τ :

1

2π

∫ ∞

−∞

eiωτ 〈 B(t− τ)A(t) 〉dτ =

Z−1∑

n,m

e−βEn < n | B | m > ×

< m | A | n > δ(En −Em

h− ω) ≡ IAB(ω) (6.21)


IAB(ω) is called the spectral density of this particular product of operators.


〈 [A(t), B(t′)]η 〉 =

∫ ∞

−∞

IAB(ω)(eβω − η)e−iω(t−t′)dω (6.22)

The Fourier representation of the Heavyside step function is

θ(t) =−i2π

∫

dωe−iωt

ω + iε, ε→ 0+ (6.23)

Exercise 6.5Prove (6.23)

With the help of (6.23), we get the Lehman spectral representation for theGreen’s functions:

G(r,a)AB =

1

2π

∫

IAB(ω)(eβω − η)

E − ω ± iεdω ≡ G±

AB (6.24)

We use now the identity

1

x± iε= P 1

x∓ iπδ(x) (6.25)

valid in the sense of the equivalence of distributions, and we obtain from (6.24) asimple relationship between the imaginary part of GAB and the spectral density:

IAB(E) =−2ImG(r)

AB(E)

eβE − η(6.26)

which is one example of the fluctuation-dissipation theorem. Since the integrandin (6.24) is proportional to Im G, that equation is one of the Kramers-Kronigreciprocal relations between the real and the imaginary part of an analyticfunction. In effect, we substitute (6.26) into (6.24) and obtain

Re G(r)AB(E) = − 1

πP∫ ∞

−∞

Im G(r)AB(ω)

E − ωdω (6.27)

Both functions in (6.24) can be considered as the two branches (in two differentRiemann sheets) of the analytic function G(E) which coincides with Gr(a) forIm E > (<)0. The branch Gr(a) is analytic in the upper (lower) half-plane.G(E) has a cut on the support of IAB(E) along the real axis [1, 2] . We cannow express the correlation function in terms of the Green’s functions.

6.2. SPECTRAL REPRESENTATION 121


〈 B(t′)A(t) 〉 = −2

∫ ∞

−∞

eiω(t−t′) Im G(r)AB(ω)η

eβω − ηdω (6.28)

Let us now prove that the retarded and advanced Green’s functions obey theKramers-Kronig relations. We already showed that they are analytic in theupper(retarded) or lower(advanced) half complex plane of the energy. Considernow the functions of the complex variable E

F± =

∮

φ±(ω;E)dω (6.29)

where

φ±(ω;E) =G±(ω)

ω −E ± iε(6.30)

and the contour for F+(F−) goes along the whole real axis, around E on aninfinitesimal semicircle in the clockwise(anti-clockwise) sense and closes alonga semicircle at infinity in the upper(lower) half plane. Let us now show thatφ±(ω;E) decreases as | ω |→ ∞ faster than 1/ | ω |. In this limit we can expandφ in inverse powers of ω:

φ±(ω;E) =G±(ω)

ω

[

1 +∞∑

n=1

(

E ± iε

ω

)n]

(6.31)

For finite E we can neglect all but the first term in (6.31). We show below that

limG±(ω) |ω→∞=cAB

ω+ O(ω−2) (6.32)

where cAB is a constant depending on the operators A,B so that the contribu-tion to (6.29) from the infinite semicircle is zero. Then, since the contour doesnot encircle any singularity of φ we get

F±(E) ≡ 0. (6.33)

If we now separate the real and imaginary parts of (6.33) we get

Re G±(E) =±π

∫ +∞

−∞

Im G±(ω)P 1

ω −Edω

Im G±(E) =∓π

∫ +∞

−∞

Re G±(ω)P 1

ω −Edω (6.34)

which are Kramers-Kronig relations.It remains to verify (6.32). To this end consider the spectral representation

(6.24). For large E we expand in inverse powers of E. Let us calculate thezeroth-order term:



∫

IAB(ω)(eβω − η)dω = cAB = 〈 AB − ηBA 〉 (6.35)

which we assume to be bounded

The whole series in (6.32) can be shown to converge if the spectrum of H isbounded. We assume then that there exists a constant 0 < W <∞ such that

| En |< W, ∀n (6.36)

Exercise 6.8Show that the series in (6.32 is bounded by a convergent geometric series if:a) the inequality (6.35) is satisfied, and b): the operators A,B have a boundedspectrum.

The functions mutually related by the Kramers-Kronig relations are a pairof mutual Hilbert transforms. We have proved therefore that the real andimaginary parts of an analytic function are Hilbert transforms of one another.Kramers-Kronig relations appear in the theory of linear response (dielectricconstant, electric conductivity, magnetic susceptibility, complex admittance ofcircuits, etc.) and in dispersion relations in the theory of scattering [1].

We mention now one useful symmetry property:


G+AB(ω) =

(

G+A†B†(−ω)

)∗(6.37)

6.3 RPA for spin 1/2 ferromagnet

Consider the Heisenbeg exchange Hamiltonian with no external field, S = 12 and

Jlm ≥ 0. Suppose we want to calculate the statistical average of 〈 Szm 〉 which,

for a translationally invariant system, is site-independent, and determines theaverage magnetization. The identity S · S = S(S + 1) yields, for spin S = 1

2 ,

S−S+ =1

2− Sz (6.38)

or

〈 S−n S

+n 〉 =

1

2− 〈 Sz

n 〉 (6.39)

Therefore

〈 Szn 〉 = σ(T ) =

1

2− 〈 S−

n S+n 〉 (6.40)

6.3. RPA FOR SPIN 1/2 FERROMAGNET 123

Since we need the equal time correlation function of S−n , S+

n , we consider theGreen’s function

G(+)lm (t− t0) ≡ 〈〈 S+

l (t);S−m(t0) 〉〉 =

−iθ(t− t0)〈 [S+l (t), S−

m(t0)] 〉 (6.41)

where η = +1 and (+) stands for retarded. Now, (6.18) reads:

ωG(+)lm (ω) =

1

πδlm〈 Sz

l 〉

− 2∑

i6=l

Jil(〈〈 S+i S

zl ;S−

m 〉〉ω − 〈〈 Szi S

+l ;S−

m 〉〉) (6.42)

Exercise 6.10Prove equation (6.42).

The “RPA” (random phase approximation) consists of neglecting the corre-lations between Sz

i and S+j for i 6= j, and substituting in (6.42) Sz

i by 〈 Szi 〉,

the statistical average, whenever it appears as argument of the Green’s function:

〈〈 S+i S

zl ;S−

m 〉〉ω ∼ 〈 Szl 〉〈〈 S+

i ;S−m 〉〉 (6.43)

The relation of (6.43) to a random phase argument can be established by re-writing the Heisenberg Hamiltonian and the equation of motion for the Green’sfunctions in k-space [3]. Let us now apply (6.43) to solve (6.42) approximately,following Bogoliubov and Tyablikov [4], for all T , in the case of a spin 1

2 fer-romagnet. Since the system is translationaly invariant by hypothesis, that isJil = J(Ri −Rl),

G(+)lm (ω) = 〈〈 S+

i ;S−m 〉〉(+)

ω =1

N

∑

k

eik·(Ri−Rm)G(+)k (ω) (6.44)

Therefore, substituting (6.43) and (6.44) into (6.42), we obtain the FT of G:


G(+)k (ω) =

σ(T )

π(ω + iε− ω(k))(6.45)

where

J(k) ≡∑

R

J(R)eik·R (6.46)

and

ω(k) = 2σ(T )[J(0) − J(k)] (6.47)

We can now apply the identity (6.40) to calculate σ(T ). First, we need the equal


time correlation function 〈 S−n (t)S+

m(t) 〉 (which is time independent). We have(6.28) at our disposal, which reduces, for t = t′ to:

〈 BA 〉 = −2

∫

dωImG(+)

AB(ω)

eβω − 1(6.48)

For the special choice B = S−i , A = S+

i ,

〈 S−i S

+i 〉 = − 2

N

∑

k

∫

dωImG(+)

k (ω)

eβω − 1(6.49)

Now,

ImG(+)k (ω) ≡ ImGk(ω + iε) =

−πδ(

ω − 2σ(T )(J(0) − J(k)) σ(T )

π(6.50)

then

〈 S−i S

+i 〉 = 2σ(T )Ψ(T ) (6.51)

where

Ψ(T ) =1

N

∑

k

1

eβωk − 1(6.52)

and

ωk = 2σ(T )(J(0) − J(k)) ≥ 0 (6.53)

Substituting〈 S−i S

+i 〉 from (6.51) into identity (6.40) we have

σ(T ) =12

1 + 2Ψ(T )(6.54)

The quantity Ψ(T ) was introduced in chapter 4, as the statistical average of thetotal number of spin deviations in a FM within the FSWA. In the present for-malism, however, it is clear that we cannot maintain that interpretation, unlessΨ → 0. We are led to the conclusion that as regards the spin deviation, theFSWA expression is obtained upon expanding the RPA formula in a series in Ψand retaining only terms up to first order. We shall see later on that the spindeviation in the FSWA is obtained from the RPA formula for general S, in thelimit S → ∞.


limT→0

Ψ(T ) = 0 (6.55)

which implies

σ(T ) |T→0→1

2(6.56)

6.3. RPA FOR SPIN 1/2 FERROMAGNET 125

If we assume that there is a finite temperature Tc, such that

Ψ(T )−→∞ , T Tc (6.57)

then we can prove that this singularity gives rise to the phase transition fromthe low T ordered FM phase to the high T disordered PM phase. Let us assumethen that as T Tc , σ → 0. From (6.52) we have in 3d,

Ψ(T ) =v08π3

∫

1stBZ

d3k1

e2βσεk − 1(6.58)

where ωk = 2σ(

J(0) − J(k))

. Since σ is small by assumption near Tc we

expand the exponential in the denominator of (6.58) in a Taylor series. Thedominant term in the expansion of Ψ is

Ψ =v0

16π3σβ

∫

d3k

εk+ O(1) (6.59)

On the other hand, from (6.54) we get:

2σΨ |TTc→ 1

2(6.60)

〈 1

f(k)〉 = F−1 =

V

8π3

∫

d3k

1 − γk(6.61)

For any f , 〈 f(k) 〉 denotes the average of f(k) over the first Brillouin zone. Inthe literature [5] the moments of

f(k) ≡ 1 − γk

are denoted asF (n) ≡ 〈 (f(k))n 〉M (6.62)

so that in particular

M 〈 1

f(k)〉 = F−1 =

v08π3

∫

d3k

1 − γk

M

(6.63)

Comparing (6.60) and (6.58) we obtain the critical temperature:

βc ≡ 1

kBTc=

2V

8π3

∫

d3k

J(0) − J(k)=

2

J(0)〈 1

1 − γk〉 (6.64)

where γk ≡ J(k)/J(0). For first n. n. range exchange we get

βcJν = 2F (−1) (6.65)

ν being the number of first n. n. in the lattice.


6.4 Comparison of RPA and MFA

We get the mean field approximation if we neglect completely the non diagonalterms in the matrix equation (6.42) for the Green-function, which means that(6.42) reduces to:

ωG(+)MFAlm (ω) =

δlm〈 Szl 〉

π+ 2

∑

i6=l

Jil〈 Szi 〉〈〈 S+

l ;S−m 〉〉 (6.66)

or

G(+)MFAlm (ω) =

δlmσ

π(ω − 2σJ(0) + iε)(6.67)

Exercise 6.13Prove that for first n. n. exchange the form (6.67) for G and Eq. (6.54) yield

σ(T ) =1

2tanh (βσνJ) (6.68)

which is the MFA result.

For σ → 0, (6.68) leads to:βcνJ = 2 (6.69)

Comparing (6.69) with (6.65),

1

kBTRPAc

=1

kBTMFAc

F (−1) (6.70)

Since F (−1) ≥ 1, TRPAc ≤ TMFA

c .By expanding Ψ near T = Tc in powers of σ one finds:

Ψ =T

4Tcσ− 1

2+ ασ

Tc

T+ . . . (6.71)

which allows to obtain the behaviour of σ as (T → Tc):

σ ∼ 1√α

√

1 − T

Tc(6.72)

This implies that the RPA yields the same critical exponent β = 1/2 for σ asthe MFA.

Exercise 6.14Verify equations (6.71) and (6.72), and check that

α =1

3F (1)F (−1)

6.5. RPA FOR ARBITRARY SPIN 127

6.5 RPA for arbitrary spin

This was first derived by R. Tahir-Kheli and D. Ter-Haar [6] and later simplifiedand extended by H. Callen [7], whose treatment we shall now follow. First, wedefine a new Green’s function which depends on a parameter a :

G(r)(l −m, t; a) = −iθ(t)〈 [ S+l (t), Bm ] 〉 (6.73)

where we define the operator Bm as

Bm = eaSzmS−

m (6.74)

and we apply again the RPA to the Green’s function which results upon calcu-lating the time derivative of (6.73) (we take h = 1):

i∂G(r)(l −m, t; a)

∂t=

δ(t)δlm〈 [ S+l (t), Bm ] 〉 +

∑

j 6=l

J(lj)

[

〈〈 Szj S

+l ;Bm 〉〉 − 〈〈 Sz

l S+j ;Bm 〉〉

]

(6.75)

We obtain, for the ( k , ω )-Fourier transform of (6.75) in RPA,

G(r)(kω; a) =〈 [S+

l , Bm] 〉ω − ω(k) + iε

(6.76)

where as in (6.45),ω(k) = 2σ(T )[J(0) − J(k)] (6.77)

For a uniform (infinite) system, σ is site independent, and so is the statisticalaverage on the right hand side of equation (6.76). We apply the general relation(6.28) between the correlation function and the retarded Green’s function, andfind:

〈 eaSzl S−

l S+l 〉 =

1

N

∑

k

〈 [ S+l , e

aSzl S−

l ] 〉eβω(k) − 1

(6.78)

Both averages appearing in (6.78) can be obtained if one calculates the function(l-independent)

Φ(a) = 〈 eaSzl 〉 (6.79)

Let us first look at the l. h. s. of (6.78). We use the identity

S−S+ = S(S + 1) − (Sz)2 + Sz

to write the average as

〈 eaSzl S−

l S+l 〉 =

S(S + 1)Φ(a) − Φ′′(a) + Φ′(a) (6.80)


Exercise 6.15Prove Eq. (6.80), where Φ′(a), Φ′′(a) are the first and second derivatives ofΦ(a).

The average on the r.h.s. of (6.78) requires the explicit calculation of

Θ(a) ≡ 〈 [ S+, eaSz

S− ] 〉 (6.81)

To this end, we use the identity

[ S+, (Sz)n ] ≡ Xn ≡ [ (Sz − 1)n − (Sz)n ]S+ (6.82)

In order to obtain (6.82), first

Exercise 6.16Prove the recursion relation:

Xn = X(n−1)Sz − (Sz)(n−1)S+ (6.83)

and then prove (6.82) by applying Peano complete induction principle of algebra.

We can now calculate Θ(a):


Θ(a) = S(S + 1)(e−a − 1)Φ(a)

+ (e−a + 1)Φ′(a) − (e−a − 1)Φ′′(a) (6.84)

We use now the function Ψ(T ) defined in (6.52), and substitute (6.80), and(6.84) into (6.78), obtaining a differential equation for Φ(a):

Φ′′(a) − Φ′(a)1 + Ψ + eaΨ

1 + Ψ − eaΨ− S(S + 1)Φ(a) = 0 (6.85)

whose complete solution requires two boundary conditions. The first one is:

Φ(0) = 1 (6.86)

The second boundary condition is obtained from the identity (spectral decom-position of Sz)

S∏

p=−S

( Sz − p ) ≡ 0 (6.87)

whose statistical average can be expressed in terms of derivatives of Φ(a) ata = 0:

(

∏

Sp=−S(

d

da− p)

)

Φ(a)| a=0 = 0 (6.88)

6.6. RPA FOR FERROMAGNETS 129

Callen [7] obtained the solution of (6.85) with the boundary conditions (6.86)and (6.88):

Φ(a) =Ψ2S+1e−Sa − (1 + Ψ)2S+1e(S+1)a

[Ψ2S+1 − (1 + Ψ)2S+1][(1 + Ψ)ea − Ψ](6.89)

Now we can calculate 〈 Sz 〉:〈 Sz 〉 = Φ′(0) =

(S − Ψ)(1 + Ψ)2S+1 + (S + 1 + Ψ)Ψ2S+1

(1 + Ψ)2S+1 − Ψ2S+1(6.90)

One verifies that for S = 1/2 this equation reduces to the result of last section.

6.6 RPA for ferromagnets

6.6.1 Paramagnetic phase

Let us now consider a uniform static external field B applied along the z axis.The initial (B → 0) longitudinal susceptibility of a FM, χzz , is defined as

V χzz = limB→0

γNσ

B, σ = 〈 Sz

n 〉 (6.91)

where V is the total volume of the system. In order to calculate the longitudinalresponse function one cannot use Gzz, since it vanishes identically, because[Sz

R, SzR′ ] ≡ 0, with R,R′ arbitrary sites on the lattice. We can use instead Gxx

or Gyy to this end, where

Gxx = 〈〈 Sxl ;Sx

m 〉〉and the corresponding definition for Gyy. If we write the perturbed Hamiltonianin the presence of the field :

H = H0 − γB∑

R

SzR (6.92)

where H0 is the Heisenberg Hamiltonian, we verify that the only change in theexpression (6.76) for the Green’s function is to add the term γB to the frequencyω(k). We call G+− the Green’s function (6.76) for a = 0 . Upon substitutingSx,y in terms of S± we find

Gxx = 1/4(G++ +G+− +G−+ +G−−)

Gyy = 1/4(G++ −G+− −G−+ +G−−) (6.93)

with a simplified notation referring to the spin operators involved. Due to totalspin conservation, G++ = G−− = 0.

If we only consider first n. n. exchange,

G+−k (ω) =

σ

π· 1

ω + iε− 2σJν(1 − γk) − h(6.94)

where h = γB.


For k = 0, the case of a uniform field, the dispersion relation ω(k) = 0 andwe have

Gxx(k = 0, ω) = Gyy(k = 0, ω) =

1

4N

∑

R

[

G+−(R,ω) +G−+(R,ω)]

(6.95)

Appealing again to total spin conservation,

[

∑

R

S+(R) , H0

]

= 0 (6.96)

and therefore we have exactly

G+−(k = 0, ω) =σ

π

1

ω − γB + iε

G−+(k = 0, ω) = −σπ

1

ω + γB + iε(6.97)

We find

Gxx(k = 0, ω) =σ

4π(

1

ω − γB− 1

ω + γB) (6.98)

And finally

V χzz = − limB→0

2πNγ2Gxx(k = 0, ω = 0) (6.99)

The self-consistency condition (6.90) involves the function Ψ defined in (6.52)We have seen that in 1 or 2 dimensions Ψ diverges at any finite T , and conse-quently σ = 0 for T > 0. In other words, a Heisenberg FM in low dimensionsat T > 0 is in the paramagnetic state.

At T = 0 however, if we let B → 0 after T → 0 we get Ψ = 0, and so for aHeisenberg FM in any dimension there is long range order at T = 0.

In the paramagnetic state, as h→ 0, σ → 0 as well, so we define the ratio

λ = limh→0

(

h

2σνJ

)

=γ2χ−1

zz

2Jν(6.100)

Then ω(k) = 2σJν(1+λ− γk) → 0 as h→ 0. Now we can take the limit h→ 0of Ψ [8] in the PM phase, and obtain:

limh→0

σΨ = S(S + 1)/3 (6.101)

which leads to an integral equation for λ:

1

N

∑

k

1

1 − γk + λ(T )=TMFA

c

T(6.102)


Solving (6.102) for each T gives us χzz , according to (6.100). In less than 3dwe follow this program for any T > 0. In 3d the spontaneous magnetizationσ 6= 0 for T < Tc, so that λ ≡ 0 in the ordered FM phase. Precisely at thecritical temperature, σ = 0, but we know this is the temperature at which thesusceptibility diverges, so that λ(TRPA

c ) = 0, and imposing λ = 0 we obtain theequation for TRPA

c

1

N

∑

k

1

1 − γk=TMFA

c

TRPAc

(6.103)

For T > TRPAc (paramagnetic phase) λ 6= 0 is the solution of the integral

equation (6.102), valid for any dimension and any spin S within RPA.At very large T TMFA

c , the r. h .s. of (6.102) is small, so that λ must belarge. We call

ηk ≡ 1 − γk

λ,

which is small if λ 1, and we expand (6.102) in a geometric series in ηk. Wedefine the small parameter α ≡ TMFA

c /T . Up to terms of order λ−2 we find

λ− 1

λ2= α (6.104)

The large solution of (6.104) for λ is

λ ≈ 1 − α

α=T − TMFA

c

TMFAc

(6.105)

which is Curie-Weiss law. Notice that extrapolating this line, λ vanishes atTMFA

c .We have seen that TRPA

c < TMFAc . The fact that the extrapolation of

the Curie-Weiss law to low T gives an intersection above TRPAc agrees with

experiments.We can also calculate the equal-time correlation function

〈 S0 · SR 〉

in the paramagnetic phase. Since in this phase the system is isotropic in theabsence of a magnetic field , we have

〈 Sx0S

xR 〉 = 〈 Sy

0SyR 〉 = 〈 Sz

0SzR 〉 (6.106)

which implies〈 S0 · SR 〉 = 3/2〈 S−

0 S+R 〉 (6.107)

The transverse correlation function on the r.h.s. above can be obtained as beforein terms of the Green’s function, by application of the fluctuation-dissipationtheorem. We obtain

〈 S0 · SR 〉 =3TkB

2νJ

1

N

∑

k

e−ik·R

1 − γk + λ(6.108)


Exercise 6.18Prove Eqs. (6.107) and (6.108).

The high T limit (6.105) is valid at any dimension, but in the low T limitfor one and two space dimensions one needs to work explicitly each case toobtain λ and χ . Let us then look at the linear chain first.

6.6.2 Linear FM chain

We write the summation in (6.102) as an integration inside the first BZ, asusual, and call TMFA

c = T1 for d = 1. The integral is standard and we get

1√

λ(λ + 2)=T1

T(6.109)

which is valid at any T > 0. From (6.100) we have

χzz =γ2

2νJλ(6.110)

which for d = 1, with ν = 2, leads to

χ1dzz =

γ2

4J(√

1 + T 2/T 21 − 1)

(6.111)

Exercise 6.19Prove Eqs. (6.109) to (6.111).

For T T1,

χ1dzz ≈ γ2T1

4J(T − T1)(6.112)

while for T → 0 we get the divergent form

χ1dzz ≈ γ2T1

T 2(6.113)

Let us now calculate the transverse equal-time correlation function, from whichwe can extract the correlation length in the PM phase:

〈 S−0 S

+R 〉 =

1

βJ

1

N

∑

k

eikR

1 − cos ka+ λ(6.114)

The integral is again standard, and the result is

〈 S−0 S

+R 〉 =

1

βJ√

λ(λ + 2)e−R/ξ (6.115)


where the correlation length for the FM chain is

aξ−1 = − log [ 1 + λ−√

λ(λ + 2) ] (6.116)


We obtain the limitsξT→0 = aT1/T (6.117)

andξT→∞ =

a

log(2T/T1)(6.118)

6.6.3 Square FM lattice

In order to obtain an explicit expression for λ from Eq. (6.102) we need tocalculate the integral

I(λ) =1

4π2

∫ π

−π

dX

∫ π

−π

dY1

1 + λ− 1/2(cosX + cosY )(6.119)

where we use the notation akx = X , aky = Y . It is very convenient to make achange of variables, defining

u = 1/2(X + Y ) , v = 1/2(X − Y ) .

The integration region in the (u, v) plane has twice the area of the BZ. Theintegral in the new variables reads, with due account of the factor 1/2 from theJacobian,

I(λ) =1

π2

∫ π

0

du

∫ π

0

dv1

1 + λ− cosu cos v(6.120)

The double integral can be reduced to a single one by use of the formula:

1

π

∫ π

0

dα

A−B cosα=

1√A2 −B2

(6.121)

Let us define the parameter

m ≡ 1

(1 + λ)2.

Then

I(λ) =2

π(1 + λ)K(m) (6.122)

where K(m) = complete elliptic integral of the first kind [9]. Finally, the equa-tion λ satisfies for the square lattice is

1

(1 + λ)π)K

(

1

(1 + λ)2

)

=T2

T(6.123)


where T2 = TMFAc for 2d (square lattice). As T → 0 the r. h. s. of (6.123)

diverges. The K function has a logarithmic singulariry near m = 1:

limm→1

K(m) =1

2log

(

16

1 −m

)

(6.124)

so that K has a logarithmic divergence at λ = 0. The singularities of both sidesof Eq. (6.123) must coincide, and we obtain for T → 0

λT→0 → 8e−(πT2/T ) (6.125)

The susceptibility at low T can be obtained now from (6.110):

χzz =γ2

64JeπT2/T (6.126)

and we verify that it diverges exponentially as T → 0.On the other hand, when T T2 , λ → ∞ and m→ 0. In this limit,

limm→0

K(m) =π

2(6.127)

so that for large T

λ =T − T2

T2(6.128)

and we recover Curie-Weiss law.Let us now calculate the correlation function in the paramagnetic phase,

which we know that for d = 2 implies T > 0. In this case we have no wayto calculate the integral in a closed form, so we look for the asymptotic limitR → ∞. We want to calculate

〈 S−0 S

+R 〉 =

1

βJν

1

N

∑

k

eik·R

1 + λ− γk(6.129)

For very large R the exponential oscillates very rapidly, leading to a cancellationof the terms for large k. Then we can limit the integral to small values of k,and expand the structure factor γk around k = 0, in a power series in the twocomponents of k,

γk = 1 − q2/4 + O(q4) , q = ak .

Although the integral over k is limited to the first BZ, we can safely extend itto ∞, since the dominant contribution comes anyway from the region aroundthe origin. Then the integral in (6.129) becomes:

1

N

∑

k

(· · ·) =1

4π2

∫ ∞

0

q dq

∫ 2π

0

dφeiq(R/a) cos φ

λ+ q2/4(6.130)

6.7. FM WITH A FINITE APPLIED FIELD 135

The exponential in (6.130) is the generating function of the Bessel functions ofthe first kind :

eim cos φ = J0(m) + 2∞∑

n=1

Jn(m)in cosnφ (6.131)

Therefore,

1

N

∑

k

(· · ·) =1

2π

∫ ∞

0

J0(qR/a)q dq

λ+ q2/4(6.132)

The integral can be found in the tables [10]:

∫ ∞

0

xJ0(ax) dx

x2 + b2= K0(ab) (6.133)

where K0(x) = modified Bessel function. For large argument the asymptoticform is [9]

K0(x)x→∞ −→√

π

2xe−x(1 + O(

1

x)) (6.134)

Finally, we can write

1

N

∑

k

(· · ·) =2

π

√

πξ

2Re−R/ξ (6.135)

with (a/ξ) = 2√λ. Therefore at low T

ξ/a =1

2√

2eπT2/2T (6.136)

which is very similar to results obtained with other methods, in particular ina reformulation of spin wave theory with a fixed number of bosons [11] and ina “large-N” expansion calculation [12]. One must keep in mind that (6.136) isan asymptotic approximation so that the smaller T becomes, the larger is thedistance at which (6.135) and (6.136) are valid, since we must have R/ξ 1for the approximations performed to be valid.

6.7 FM with a finite applied field

If the applied field h is large, that is if βh 1, the expression (6.52) for Ψsimplifies:

Ψ ≈ 1

N

∑

k

e−βω(k) (6.137)

where

ω(k) = h+ Jνσ(1 − γk) (6.138)


Assume that we are dealing with a simple hypercubic lattice in d dimensions.Then the structure factor is

γk =1

d

d∑

m=1

cosakm (6.139)

The integrand in (6.137) can be decomposed as a product of identical functionsof the d independent coordinates. Each integral is of the form

∫ π

0

dt e−A cos t = I0(A) (6.140)

where I0 is a modified Bessel function [9] and A = 2σJ/kBT . Therefore,

Ψ ≈ e−(h+Jνσ)/kBT

(

I0(2σJ

kBT)

)d

= e−h/kBT

(

e−2σJ/kBT I0(2σJ

kBT)

)d

(6.141)

since ν = 2d in a hypercubic lattice. If kBT J , we can use the asymptoticform [9]

e−zI0(z) →1√2πz

(6.142)

Therefore, in the case J kBT and h kBT we obtain

Ψ ∼ e−h/kBT

(

kBT

4πσJ

)d/2

(6.143)

Since by assumption the field is large and the temperature low, σ will be almostsaturated. This means that Ψ 1, and we can expand σ in (6.90) for smallvalues of Ψ, that is [7]

σ = S − Ψ + (2S + 1)Ψ2S+1 − · · · (6.144)

and we obtain finally

σ ≈ S − e−h/kBT

(

kBT

4πσJ

)d/2

(6.145)

6.8 RPA for antiferromagnet

Let us now turn to a two-sublattice antiferromagnet with Hamiltonian

H =∑

<ab>

JabSa · Sb − ha

∑

a

Sza + ha

∑

b

Szb (6.146)

where the negative sign of the exchange interactions has been taken into ac-count. We include, as in Chap. 3, a staggered anisotropy field Ba and we callha = γBa. Points in the ↑ (↓) sublattice are called a (b).

6.8. RPA FOR ANTIFERROMAGNET 137

6.8.1 Spin 1/2 AFM

We define now retarded Green’s functions as for the FM

G11(Ra −R0, t) = 〈〈 S+a ;S−

0 〉〉G21(Rb −R0, t) = 〈〈 S+

b ;S−0 〉〉 (6.147)

where R0 belongs to the up sublattice. We call σ = 〈 Sza 〉 = −〈 Sz

b 〉. Now theFourier transformed equations of motion for the pair of Green’s functions are

(ω − h− 2σJ0)G11 + σJkG21 =σ

π(ω + h+ 2σJ0)G21 − σJkG11 = 0 (6.148)

where we dropped the subindex a in h and the tilde on Jk to simplify thenotation. Let us now define the quantities

ε1k = h+ 2σ(J0 − Jk)

ε2k = h+ 2σ(J0 + Jk)

ε2k = ε1kε2k (6.149)

Then with a little algebra one finds

G11 =σ

4πεk

(

2εk + ε1k + ε2k

ω − εk− ε1k + ε2k − 2εk

ω + εk

)

G21 = −σ2Jk

2πεk

(

1

ω − εk− 1

ω + εk

)

(6.150)

The poles of the Green’s functions are ω = ±εk. To see what spectrum thisyields, let us specialize to the case with only first n. n. interactions, whereJk = Jνγk. Then,

εk =√

(h+ 2Jνσ)2 − (2Jνσγk)2 (6.151)

In the absence of anisotropy we find for small k a linear dispersion relation

εk = sk (6.152)

Exercise 6.21Show that for the square lattice in 2d we find

s = 8Jσ(T )a/√

2

where a = atomic lattice constant.

This is a dispersion relation typical of a massless boson excitation, like pho-tons or acoustic phonons. We explicitly indicate the T dependence of the AFmagnon velocity through σ. As T increases, the velocity decreases, and it van-ishes with σ at TN . Although some renormalization of s with temperature is


to be expected, its scaling with the magnetization, as found above, is a directconsequence of the RPA renormalization of the magnon excitation energy in(6.151). If there is an anisotropy staggered field h the dispersion relation has agap at k = 0. For small k we obtain from (6.151) the expansion

εk = ∆ +ω2

e

2∆A(ka)2 (6.153)

where ∆2 = ω2a + 2ωaωe and we are using the notation of Chap. 3: ωa = h,

ωe = Jνσ. The geometric factor A in (6.153) depends on the lattice, and forthe square lattice it is 1/2.

6.8.2 Arbitrary spin AFM

Let us now extend the RPA for the AFM to arbitrary values of S. To this endwe define as for the FM a new Green’s function, which depends on a parameterv as:

G11(Rn, ω; v) = 〈〈 S+n ; evSz

0S−0 〉〉ω (6.154)

and the quantityΘ(v) = 〈 [S+

n , evSz

0S−0 ] 〉 (6.155)

In particular, Θ(0) = 2〈 Sz0 〉. Let us also define

U(v) = 〈 evSz0S−

0 S+0 〉 (6.156)

In the RPA we obtain for the imaginary part of the retarded Fourier transformedGreen’s function :

ImG11(k, ω + iε; v) =Θ(v)

4εk[−(2εk + ε1k + ε2k)δ(ω − εk) +

(ε1k + ε2k − 2εk)δ(ω + εk)] (6.157)

The fluctuation-dissipation theorem relates U and Θ through

U(v) =1

4N

∑

k

Θ(v)

4εk

[

2εk + ε1k + ε2k

eβεk − 1+

2εk − ε1k − ε2k

e−βεk − 1

]

(6.158)

which can also be written as

U(v) =

(

1

2N

∑

k

ε1k + ε2k

2εkcoth(βεk/2)− 1/2

)

Θ(v) (6.159)

In the sums above N is the number of sites in each magnetic sublattice, namelyhalf the total number of spins in the system. Let us call

Φ ≡ 1

2N

∑

k

ε1k + ε2k

2εkcoth(βεk/2) (6.160)


In order to make contact with Callen’s formalism for the RPA we write

U(v) = ΨΘ(v) (6.161)

and define Φ ≡ Ψ+1/2. Since we only need local averages on a given sublattice,we recognize that the problem is formally identical to the one for the FM, sothat the general solution for the local magnetization is

〈 Sz0 〉 = (S + 1/2)

(Φ + 1/2)2S+1 + (Φ − 1/2)2S+1

(Φ + 1/2)2S+1 − (Φ − 1/2)2S+1)− Φ (6.162)

which is the form (6.90) takes in terms of Φ. Let us now quote the expansionsof (6.162) in the limits of Φ very small or very large. For Φ → 0 we get [7]

σ = S − Φ + (2S + 1)Φ2S+1 − (2S + 1)2Φ2S+2 + O(Φ2S+3) (6.163)

while for Φ → ∞ (Ψ → ∞)

Φσ → S(S + 1)/3 (6.164)

6.8.3 Zero-point spin deviation

In 3d an AFM at T = 0 has a finite sublattice spin deviation, due to the zero-point fluctuations of the sublattice magnetization which we already described inChap. 4 within the FSWA. From (6.160) we find that as β → ∞, Φ → constant,and this constant depends on the dimensionality and on the parameters of thesystem. If we restrict ourselves to first n. n. interactions, we find for a squarelattice [8] (see section 4.3.3):

limT→0

Φ = 0.697 (6.165)

For S = 1/2 this yields σ = 0.358. For S → ∞ instead

σ → S − Ψ = S − 0.197 , (6.166)

which agrees with the spin wave theory results [15], Eqs.(4.51) and (4.83) Infact expression (6.166) is obtained within the FSWA for any spin, which showsthat in this respect the FSWA is the S → ∞ limit of the RPA.

In 1d, Ψ → ∞ in the absence of anisotropy, which in the FSWA implies thatσ diverges. In the RPA this implies on the contrary that σ → 0 when h → 0.Both results agree in the sense that there is no LRO in 1d in the absence ofanisotropy. We remind that in chapter 4 we defined the anisotropy parameterα = h/(JSν), in terms of which

Ψ(1d) = − 1

2S

(

1 + log2α

π

)

(6.167)

The weak divergence of Ψ implies that in RPA even a very small anisotropyleads to a finite σ at T = 0. The absence of LRO in the isotropic 1d AFMwas rigourously proven in 1931 by Hans Bethe [13], who obtained a completeanalytic solution for the ground state of the isotropic Heisenberg AFM chainwith first n. n., known as the Bethe Ansatz theory.

In 3d, Ψ is finite. Results for some lattices were quoted in chapter 4.


6.8.4 Correlation length

By applying the fluctuation-dissipation theorem, we have

〈 S−(0)S+(R) 〉 = −2

∫

dωIm〈〈 S+(R);S−(0) 〉〉ω+iεN(ω) (6.168)

where

N(x) ≡ 1

eβx − 1

and

〈〈 S+(Rα);S−(0) 〉〉ω+iε =1

N

∑

k

Gα1(k, ω + iε)e−ik·Rα

Here α = 1(2) if Rα is the position vector of a spin in sublattice ↑ (↓).If the lattice has inversion symmetry then G(−k, ω) = G(k, ω), so that we

can write (6.168) as

〈 S−(0)S+(Rα) 〉 = −2

∫

e−ik·RαN(ω)ImGα1(k, ω + iε)dω (6.169)

Now we take the imaginary part of (6.150) and substitute it in the last equation.For points on the same (up) sublattice we get

〈 S−(0)S+(Ra) 〉 = σ∆(R) +2σ

N

∑

k

e−ik·Raε1k + ε2k

4εkcothβεk/2 (6.170)

with ∆R = 0, R 6= 0 ; ∆(0) = 1.For points on different sublattices,

〈 S−(0)S+(Rb) 〉 = − 1

N

∑ σ2Jνγk

4εkcoth (βεk/2) e−ik·Rb (6.171)

We shall now look at these formulae in the paramagnetic (PM) phase, so we takethe limit σ → 0. We remark that since we maintain an anisotropy staggeredapplied field h, we can still consider the system as infinitesimally polarized witha two-sublattice magnetic structure, and get a finite λ in the limit h → 0. Wefind:

〈 S−(0)S+(Ra) 〉 =1

2NβJν

∑

k

e−ik·Ra

[

1

1 + λ− γk+

1

1 + λ+ γk

]

(6.172)

and

〈 S−(0)S+(Rb) 〉 =−1

2NβJν

∑

k

e−ik·Rb

[

1

1 + λ− γk− 1

1 + λ+ γk

]

(6.173)

One important simplification arises from a property of the structure factor γk

in the case of first n. n. interactions. In this case it is possible to find a vectorQ with the property

Q · (Rα −Rβ) = 2mπ


if α, β belong to the same sublattice, and

Q · (Rα −Rβ) = (2m+ 1)π

if they belong to different sublattices. We then have

γk+Q = −γk (6.174)

For the vector Q in the square lattice we have the choices

Q = (π/a) (±1,±1) .

Any vector in this set carries the first magnetic BZ into the first atomic one, andfor each of these translations there is a change of sign of γk. Taking advantageof this symmetry of the structure factor we can write the sum over the magneticBZ as a sum over the atomic one, and for 0 < T T2 we find for the correlatorexactly the same expression as in the case of a FM, except that the sign changesas we change from the ↑ to the ↓ sublattice:

〈 S−(0)S+(Ra) 〉 =1

2NβJν

∑

k∈ BZat

e−ik·Ra1

1 + λ− γk(6.175)

and

〈 S−(0)S+(Rb) 〉 = − 1

2NβJν

∑

k∈ BZat

e−ik·Rb1

1 + λ− γk(6.176)

Exercise 6.22Prove Eqs. (6.175) and (6.176).

The calculation of the Neel temperature TN can be done now by simply takingformula (6.103) for the Curie temperature Tc of a FM and remembering thatthe integral in k space must be extended over the atomic BZ, which we indicateby a prime on the summation simbol. In the absence of anisotropy, we have:

1

N

∑

k

′ 1

1 − γk=S(S + 1)νJ

3kBTN(6.177)

Regarding the correlation length, if T > 0 in 1d we have the same expresion asfor the FM chain, except that we keep track of the oscillating sign:

〈 S0 · SR 〉 = (−1)νS(S + 1)e−R/ξ (6.178)

where ν = 1(0) for different (the same) sublattice points at distance R, andwhere ξ is the same as in (6.116). The corresponding result for the squarelattice is

〈 S0 · SR 〉 = (−1)ν 3kBT

2J

(

ξ

2πR

)1/2

e−R/ξ , (6.179)


with ξ from (6.136). A similar result is obtained in the two-loop approximationof Chakravarty, Halperin and Nelson [14]. Let us look now at the T → 0 limitfor the AFM chain. In this limit, cothβεk/2 → 1, so that Φ is

Φ =1

N

∑

k

1√

1−m2γ2k

(6.180)

with m ≡ (1 + λ)−1. The resulting integral is proportional to the completeelliptic integral of the first kind [10] and for λ → 0, which is the limit for T → 0,we find

Φ → 2

πlog

4

λ(6.181)

On the other hand we know that as Φ diverges we have Φσ → S(S+1)/3. Then

λ = 4 exp−(

S(S + 1)π

6σ

)

(6.182)

From the definition of λ we have h = λσνJ , so as the staggered field h→ 0we also have σ → 0 and then λ → 0 as T → 0. Finally,

χ =γ2

2νJλ→ ∞

that is, the initial staggered susceptibility diverges as T → 0.According to (6.117) ξ diverges in 1d for the FM chain as T → 0, so in the

AFM we have〈 S0 · Sna 〉 = (−1)nS(S + 1) (6.183)

where a is the atomic lattice constant. This result is clearly wrong at longdistances, since it implies the existence of long range order. A calculation forthe S = 1/2 AFM chain within the Green’s function formalism was publishedby Kondo and Jamaji [16] in which a RPA-type decoupling is performed in thenext order of the hierarchy of equations, thus allowing to incorporate correctlythe vanishing of σ, and to obtain self-consistently the correlators cn ≡ 〈 Sz

1Szn 〉.

The results for the correlators cn for n = 1, 2 agree reasonably well both with theexact result [18], c1 = −0.59086 and with the numerical calculation on a chainof 11 spins performed by Bonner and Fisher [19], which yields c2 = 0.25407.The agreement is much better for the first nearest neighbours. One suspectsthat in order to obtain reasonable results for cn with the decoupling approach ,one must go down to the n-th level of the hierarchy of equations. At any rate, c1is just what is needed for calculating the specific heat, as we already mentionedin Chap. 3. The agreement of the specific heat calculated with the Green’sfunction decoupling in the second level with the numerical estimates based onfinite size scaling is fairly good.

In 3d there is a finite transition temperature TN , so that the correlationlength is finite for T > TN . In this phase λ is finite and we can use (6.108), withthe sum over k extended inside the atomic BZ. If we are interested in the longdistance behaviour of the correlator it is enough to keep up to the quadratic

6.9. RPA SUSCEPTIBILITY OF AFM 143

terms in the expansion of γk for small k in the denominator of (6.108). Then,transforming the sum into an integral we have

〈 S0 · SR 〉 =3kBT

(2π)3νJ

∫

BZ

a3d3ke−ik·R

Aa2k2 + λ(6.184)

where a =lattice constant and A is a numerical coefficient which depends onthe lattice. For large R we can extend the integral to the whole volume, whicheventually tends to infinite, and then (6.184) is just the Yukawa potential:

〈 S0 · SR 〉 =ξe−R/ξ

4πAR(6.185)

whereξ = a

√

A/λ (6.186)

Exercise 6.23 Prove (6.185) and (6.186).

Observe that ξ is inversely proportional to√χ. At high T then

ξ ∼ 1√

T − TMFAc

while as T TN , ξ diverges as (T −TN)−1/2, and it is infinite all the way downto T = 0 in the ordered phase. In fact in the AFM phase the correlator (6.184)∼ R−2.

6.9 RPA susceptibility of AFM

We study now the response of an AFM to a uniform static external field. Is isconvenient to include some anisotropy in the model from the start, and we choosenow an exchange anisotropy term, which originates from the combined effect ofthe spin-orbit interaction of the un-paired electrons of the ions and the crystal-field. The resulting effect in the effective spin Hamiltonian is an anisotropicexchange interaction. We assume that the exchange zz term is different fromthe others, and restrict ourselves to first n. n. exchange interactions amonga (↑) and b (↓) spins:

H =∑

a,b

(JabSa · Sb +KabSzaS

zb

−γB(∑

a

Sza + Sz

b ) (6.187)

We shall calculate the generalized retarded Green’s functions defined inEq. (6.154), where we introduced the operator

B0(v) ≡ evSz0


where as before the origin is taken at an up spin site (sublattice a) and we alsodefine

Bb0 ≡ evSzb0

where b0 denotes the origin of the b (down) sublattice. As before, indices 1(2)refer to ↑ (↓) sublattices. The retarded Green’s functions satisfy the RPAequations:

(E − γB)G11(a,E; v) =Θa(v)

2πδRa,0 + 2

∑

b

JabσaG21(b, E; v)

− 2(Jab +Kab)σbG11(a,E; v) (6.188)

and

(E − γB)G21(b, E; v) = 2∑

a

JabσbG11(a,E; v)

− 2(Jab +Kab)σaG21(b, E; v) (6.189)

with a simplified notation in which a(b) substitutes R(a,b), and all sites havecoordinates referred to the origin in the a sublattice.

Let us remark that in the presence of a uniform external field we do not haveany more the time-reversal symmetry σb = −σa. We notice that this symmetryis apparent in the self-consistency equation (6.162), since one verifies that thereσ(Φ) = −σ(−Φ). We must now retain explicitly the reference to the sublattice.In particular, we generalize the definition of Θ(v) in Eq. (6.155) and considertwo different functions Θ(a,b)(v). Fourier transforming to k space we get:

(E − γB)G11(k,E; v) =Θa(v)

2π+ 2JkσaG21(k,E; v)

− 2(J0 +K0)σbG11(k,E; v) (6.190)

and

(E − γB)G21(k,E; v) = 2JkσbG11(k,E; v)

− 2(J0 +K0)σaG21(k,E; v) (6.191)

For first nearest neighbours Jk = Jνγk and the roots of the secular determinantof the system of equations above are

E±k − ω0 = (J +K)ν(σa + σb) ±

√

∆k (6.192)

where ω0 = γB and the discriminant is

∆k = (J +K)2ν2(σa + σb)2 − 4ν2σaσb

[

(J +K)2 − J2γ2k

]

(6.193)

We see that in the absence of a magnetic field we have

E±k = ±E0 = ±2νJσ

√

(1 + α)2 − γ2k (6.194)

6.9. RPA SUSCEPTIBILITY OF AFM 145

and we recover exactly the dispersion relation (6.151).Notice that the anisotropy parameter α plays exactly the same role as the

parameter λ did for the FM.If we follow now the same procedure as before, we find

G11(k,E) =σa

π

[

1

E − E++

1

E −E−

]

+2(J +K)νσa(σa − σb)

π(E+ −E−)

[

1

E −E+− 1

E −E−

]

(6.195)

and

G21(k,E) =2(J +K)νσ2

a

π(E+ −E−)

[

1

E −E+− 1

E −E−

]

(6.196)

The self-consistency equation (6.158) relates Θ(v) with the function U(v) de-fined in (6.156). The only change we need to introduce now is to identify thesublattice, so we write

Θa(v)Ψa = −21

N

∑

k

∫

ImG11(k,E + iε; v))N(E)dE (6.197)

Then

Ψa =1

2N

∑

k

[

1

eβE+ − 1+

1

eβE− − 1

]

+1

2N

∑

k

2(J +K)ν(σa − σb)

π(E+ −E−)

[

1

eβE+ − 1− 1

eβE− − 1

]

(6.198)

while for the b sublattice one should simply interchange a and b above. Thepoles E± are invariant under that interchange, while the sign of the secondterm changes. Therefore, with the convenient change of notation a(b) → +(−),we have

Ψ± =1

2N

∑

k

[

1

eβE+ − 1+

1

eβE− − 1

]

± 1

2N

∑

k

2(J +K)ν(σ+ − σ−)

π(E+ −E−)

[

1

eβE+ − 1− 1

eβE− − 1)

]

(6.199)

We make the assumption that under a small field the linear response is the samefor both sublattices, so we write

σ± = 〈 Sz± 〉 = ±σ0 + αB (6.200)

where σ0 is the spontaneous sublattice spin polarization.The coefficient α is proportional to the susceptibility:

χ‖ = γρα (6.201)


where ρ = N/V is the volume concentration of the magnetic spins in the sample.We now recall that σ can be expressed in terms of Ψ through Eq. (6.162), whereΦ = Ψ + 1/2, and this is true for both sublattices, so that we express α, using(6.200), as

α =1

2

∑

m=±

∂〈 Szm 〉

∂Ψm

(

∂Ψm

∂B

)

B=0

(6.202)


α = limB→0

1

2

∑

m=±

[

(

(2S + 1)Ψm(1 + Ψm)S

(1 + Ψm)2S+1 − Ψ2S+1m

)2

− 1

]

(

∂Ψm

∂B

)

(6.203)

In the PM phase (T > TN ) Ψ diverges as B → 0, but (6.203) has a finitelimit:

Exercise 6.25Prove that the limit (6.203) yields for α−1 the expression

α−1 =3kBT

S(S + 1)N

∑

k

γ − J0α

(γ − J0α)2 − (J0αγk)2(6.204)

where J0 = 2νJ .If we choose T = TN the last equation must be identical to (6.177), which

requires that α(TN ) = γ/2J0, and also implies that

γ − J0α(TN ) > 0

for consistency. We found in the MFA that χ‖(TN) is the absolute maximum ofχ‖(T ), so that the inequality above is satisfied for all T in the MFA. If this bealso true in the RPA, (6.201) and ( 6.204) would determine χ‖ in the PM phasewithin this approximation. We shall assume the inequality above to be valid.We ought now to obtain χ‖ in the RPA for T < TN . Before that we mustmake a digression, since in the absence of anisotropy the AFM is unstable uponapplication of a longitudinal uniform field.

6.10 Spin-flop transition

We remind that this transition is the result of the instability of the AFM phaseunder the action of a static uniform field parallel to the sublattice magnetization,a phenomenon we studied within the MFA in Chap. 3. From the dynamicalpoint of view one expects that a soft magnon be responsible for this instability.

6.10. SPIN-FLOP TRANSITION 147

Therefore let us study the dispersion relation when both anisotropy and externalfield are present. The frequencies of both branches are

E± = γB − (J +K)ν(σa + σb) ±√

∆k (6.205)

where∆k = [(J +K)ν(σa + σb)]

2 − (2Jνγk)2σaσb (6.206)

The vanishing of the lower branch will entail the divergence of Φ and in conse-quence an instability of the AFM phase, since then the LRO parameter σ wouldvanish. From (6.200) we get

σa + σb = 2αB

σaσb = B2α2 − σ20 (6.207)

Then∆k = 4(J +K)2ν2α2B2 − 4(Jνγk)2(α2B2 − σ2

0) (6.208)

Since B is a small perturbation, α2B2 − σ20 ≤ 0 and we have

∆k = ∆0 − (2Jν)2(1 − γ2k)(σ2

0 − α2B2) ≤ ∆0 (6.209)

Therefore the critical condition for the vanishing of the lower (−) branch is

BSF [ γ − 2(J +K)να ] =√

∆0 (6.210)

If the anisotropy is small ( K J) we can expand (6.210) and retain only thefirst order term in K/J . One finally obtains for the critical field for spin-flop:

B2SF ' 8Kν2Jσ2

0

γ2 − 4γαJν(6.211)

Now, γα = χ‖/ρ. We recall the MFA result

χMFA⊥ =

γ2

4Jνρ

Then we find

B2SF =

2Kνσ20ρ

χMFA⊥ − χ‖

(6.212)

We see that in the absence of anisotropy (K = 0) an infinitesimal field de-stabilizes the AFM phase. We have seen in Chap. 3 that the MFA predicts thatin the new stable configuration the sublattices align approximately perpendic-ular to the field. When K = 0 both magnon branches are degenerate whenB = 0, a result we already obtained with the FSWA in Chap. 4. The effectof the field is to break this degeneracy, which is the result of the time reversalsymmetry of the Hamiltonian. The energy of the lower branch decreases asB increases, and eventually becomes negative. A negative excitation energy isnaturally a symptom of instability of the assumed ground state configurationand as a result the system goes into a new ground state and the excitationsmust be redefined accordingly. In the spin-flop phase we still have two orderedsublattices, and the excitations are spin waves, which were obtained by Wangand Callaway [20].


6.11 χ‖ at low T

If the Hamiltonian has no anisotropy terms the system becomes completelyisotropic at TN , so that χ‖(TN) = χ⊥(TN ). If T < TN we have an AFM phasewith broken symmetry, and χ‖(T ) 6= χ⊥(T ). We show now that

limT→0

χ‖(T ) = 0

Let us consider a small applied field along the spin quantization axis at low T .Then the poles of the Green’s function are

E±(k) = ±Ek + δ (6.213)

where Ek are the unperturbed poles in the absence of field and

δ = B[γ − 2(J +K)να] (6.214)

According to Eq. (6.201), αγ = χ‖/ρ. We are interested in the zero-fieldsusceptibility, so that eventually δ → 0. The system has an exchange anisotropyK > 0, so that it will not undergo the transition to the spin-flop phase with aninfinitesimal field. If βJ 1 the spontaneous sublattice spin polarization σ0

approaches the zero T limit, so Ψ ≈ 0 and we have from (6.163)

(

∂σ±∂Ψ±

)

Ψ→0

→ −1 (6.215)

Then

χ‖(T ) = −γ4

(

∂(Ψ+ + Ψ−)

∂B

)

B=0

(6.216)

Exercise 6.26Prove Eq. (6.216) and show that from (6.199) we get:

χ‖(T ) =β

4

[

γ2 − 4(J +K)νχ‖(T )

ρ

]

A(β) (6.217)

where

A(β) =1

N

∑

k

eβEk

(eβEk − 1)2(6.218)

The second term in the square brackets on the r. h. s. of Eq. (6.217) can beneglected since χ‖(T ) is very small. The sum in Eq. (6.218) can be simplifiedbecause βJ 1 and we can use the small k approximation Ek ≈ sk becausecontributions from large values of k will be exponentially small in (6.218). AfterB → 0 we can let the anisotropy vanish, and this is why we have used the lineardispersion relation which is obtained for K = 0. Then

χ‖(T ) =ργ2a3k2

BT2

24s3(6.219)

6.12. TRANSVERSE SUSCEPTIBILITY 149

The magnon velocity depends on J and on the geometry of the lattice. Inparticular for the b. c. c. atomic lattice,

χ‖(T ) =γ2ρ

3S3J

(

kBT

16SJ

)2

(6.220)

With anisotropy the dispersion relation has a gap at k = 0 and(

A(β) , χ‖(T ))

vanish exponentially for T → 0.

6.12 Transverse susceptibility

We assume now that the field is perpendicular to the sublattice magnetization,say in the x direction. One fairly direct way to obtain the response of the systemis through the free energy. The Hamiltonian in the presence of the field is

H = H0 −M ·B (6.221)

We can consider the Helmholtz free energy as a function of the total magneticmoment, the temperature and the magnetic field. Then the zero-field magneticmoment in the x direction is

Mx = −(

∂F

∂Bx

)

B=0

(6.222)

and for the susceptibility we have

χxx =1

V

(

∂Mx

∂Bx

)

B=0

= − 1

V

(

∂2F

∂B2x

)

B=0

(6.223)

From the definition of F we find

χxx =

(

β

V

)

[ 〈 M2x 〉 − (〈 Mx 〉)2 ] (6.224)

where the statistical averages are evaluated in the absence of the external field.Because of this, Mx averages to zero, since we assumed that the spins werealigned in the ±z directions, and we are left with

χ⊥ = βγ2ρ1

2N

∑

l,m

〈 S−l S

+m 〉 (6.225)

The expression above is valid both in the AF and the PM phases. In the PMphase we must show that χ⊥ coincides with χ‖ as calculated before, due toisotropy. Eq. (6.224) is again the fluctuation-dissipation theorem, in this casein the static limit.

We remind that for any pair of operators A,B,

∑

R

〈 ARB0 〉 = − 2

N

∑

R, k

e−ik·R

∫

Im〈〈 B0;AR 〉〉k,E+iεN(E)dE (6.226)


and we recall that∑

R

e−ik·R = N∆k,0 (6.227)

where ∆k,0 is Kronecker’s delta. We calculate now the sums in (6.225) directlyby summing (6.170) and (6.171):

∑

Rα

〈 S−RαS+

0 〉 = −2∑

k

∆k,0

∫

ImGα1(k,E + iε)N(E)dE (6.228)

Let us consider the AFM phase. For points on the same sublattice

∑

R+

〈 S−R+S+

0 〉 =∑

k

∆k,0

2νβ

(J +K)

(J +K)2 − J2γ2k

(6.229)

and for points on different sublattices

∑

R−

〈 S−R−S+

0 〉 = −∑

k

∆k,0

2νβ

Jγk

(J +K)2 − J2γ2k

(6.230)

Adding up (6.229) and (6.230) and substituting in (6.225) we find:

χ⊥ =γ2ρ

2ν(2J +K)(6.231)

which coincides with the MFA result and is temperature independent.

6.13 Single-site anisotropy

We already found this anisotropy in Chap. 1, as one of the effects of the spin-orbit interaction in the presence of a crystal field. Let us now study the thermo-dynamic and the dynamic effects of a term in the Hamiltonian of the uniaxialanisotropy form

Ha = −1

2D∑

i

(Szi )2 (6.232)

We assume D > 0 (otherwise spins will align predominantly perpendicular tothe z axis, in the x, y plane, which is why the case D < 0 is called the XYmodel). Let us consider the terms that result upon the commutation of S+

n

with the anisotropy term in the Hamiltonian. We call the resulting operator

−A(2)n :

−A(2)n = [ S+

n , (Szn)2 ] = −(Sz

nS+n + S+

n Szn) (6.233)

We cannot apply the argument that led to the RPA, since both operators onthe r. h. s. of (6.233) refer to the same site and are strongly correlated. One

6.13. SINGLE-SITE ANISOTROPY 151

can instead obtain the equation of motion of the new Green’s function which

contains the operator A(2)n . Devlin [21] defines three recurrent sets of operators:

A(1)n = S+

n

A(ν)n = −[A(ν−1)

n , (Szn)2] , 2S ≥ ν ≥ 2 (6.234)

B(ν)n = [A(ν)

n , S+n ]

C(ν)n = [A(ν)

n , S−n ] (6.235)

Exercise 6.27Prove that C

(ν)n =polynomial in Sz

n of order ν, which contains only even (odd)powers if ν is even (odd).

Exercise 6.28Prove that ∀ν,

[ A(ν)n , Sz

n ] = −A(ν)n (6.236)

The last step in this preparatory background is to show that there are only

2S − 1 linearly independent operators A(ν)n . This comes about because the ma-

trix S+ connects only a state with azimuthal quantum number m with that withm+ 1. In the manifold of spin S, where the spin matrices are of order 2S + 1,this operator contains 2S non-zero parameters and it is therefore a linear com-bination of 2S linearly independent matrices out of the (2S + 1)2 dimensionalbasis set that spans the whole manifold of matrices. On the other hand, onecan prove that

A(ν)n = (2Sz

n − 1)A(ν−1)n (6.237)


Since all powers of Sz are diagonal, all matrices A(ν)n have the same struc-

ture and they are therefore different linear combinations of the same 2S basismatrices. This implies that there cannot be more than 2S linearly independentmatrices of this particular structure. As a consequence we can write, with achoice of sign convenient for what follows,

A2S+1n = −

2S∑

h=1

ahAhn (6.238)


Exercise 6.30Show that for S = 1 , A3 = A.

We consider now 2S Green’s functions (we include an external field):

〈〈 A(ν)l ;S−

m 〉〉, ν = (1, · · · , 2S)

and find the Fourier transformed equations of motion for ν ≤ 2S − 1:

(E − γB)〈〈 A(ν)l ;S−

m 〉〉E =δlm2π

〈 C(ν)l 〉 +

D

2〈〈 A(ν+1)

l ;S−m 〉〉E

+∑

j

(

2J〈〈 A(ν)l Sz

j ;S−m 〉〉E

− J〈〈 B(ν)l S−

j ;S−m 〉〉E

)

− J∑

j

〈〈 C(ν)l S+

j ;S−m 〉〉E (6.239)

The equation for ν = 2S becomes:

(E − γB)〈〈 A(2S)l ;S−

m 〉〉E =δlm2π

〈 C(2S)l 〉 − D

2

2S∑

h=1

〈〈A(h)l ;S−

m 〉〉E

+∑

j

( 2J〈〈 A(2S)l Sz

j ;S−m 〉〉E

− J〈〈 B(2S)l S−

j ;S−m 〉〉E

− J〈〈 C(2S)l S+

j ;S−m 〉〉E ) (6.240)

Exercise 6.31Verify (6.239) and (6.240).

We can now perform the approximations analogous to the usual RPA:

〈〈 A(ν)l Sz

j ;S−m 〉〉E ≈ 〈 Sz

j 〉〈〈 A(ν)l ;S−

m 〉〉E〈〈 B(ν)

l S−j ;S−

m 〉〉E ≈ 0

〈〈 C(ν)l S+

j ;S−m 〉〉E ≈ 〈 C(ν)

l 〉〈〈 S+j ;S−

m 〉〉E (6.241)

Let us simplify the notation, and call 〈 C(ν)l 〉 = Zν , which is site independent.

For spin S Devlin’s equations are [21]:For ν ≤ 2S − 1:

Zν

2π= (E − γB − 2σ(J0)G

(ν)(k,E) + ZνJkG(1)(k,E)

− D

2G(2ν+1)(k,E) (6.242)

6.14. DYNAMIC LINEAR RESPONSE 153

For ν = 2S:

Z2S

2π= (E − γB − 2σJ0)G

(2S)(k,E) + Z2SJkG(1)(k,E)

− D

2

2S∑

h=1

ahG(h)(k,E) (6.243)

For a given S we obtain a magnon spectrum with 2S branches. D/J → ∞ is theMFA limit. For the opposite limit D/J → 0 Devlin’s results are similar to thoseof Lines [22]. We refer the reader to Devlin’s papers for details of calculationsfor FM and AFM systems and for comparison with other decoupling schemesand the MFA [22, 23, 24, 25].

6.14 Dynamic linear response

We shall now consider the effect of a time dependent external perturbation on aspin system, and show that the linear dynamic susceptibility can be expressedin terms of the retarded Green’s functions. The central fact to stress is that inthe linear approximation the Green’s functions involved are those of the systemin thermodynamic equilibrium.

To the same order of approximation, the power transferred from the systemto the external probe, or in other words, the rate of energy dissipated by oursystem, turns out to be proportional to the imaginary part of the dynamic sus-ceptibility. This is the reason for the expresssion fluctuation-dissipation theoremas the general relation of proportionality between the power absorbed from anexternal probe and the fluctuations of the relevant variables in the unperturbedsystem. A relation of this form is a particular example of the general Kuboformula [26].

First of all, let us briefly revise the basic ideas of linear response theory. Thecomplete Hamiltonian of a perturbed system is

H(t) = H0 + V (t) (6.244)

H0 is the Hamiltonian of the unperturbed system, which is not time dependent.The external probe couples to this system through the time-dependent potentialV (t).

The density matrix, in the Schrodinger picture, satisfies the linear differencialequation:

ih∂ρs

∂t= [ H(t), ρs] (6.245)

Transforming (6.245) to the interaction picture, each operator is transformedas:

A(t) = U0(t)As(t)U †

0 (t) (6.246)

where U0(t) ≡ eiH0t/h. In Schrodinger’s picture (SP) dynamic operators (cor-responding to observables of the unperturbed system) are not time-dependent.


In the interaction picture, ρ satisfies:

ih∂ρ(t)

∂t= [ V (t), ρ(t) ] (6.247)

where V (t) = U0(t)Vs(t)U †

0 (t)


In the case of a magnetic system driven by an external field the interactionterm in the Hamiltonian is:

V (t) = −γ∑

l

B0(Rl, t) · Sl(t) (6.248)

with Sl(t) in the interation picture. We assume that V (t) is “small”, whichimplies that in Eq. (6.247) we can decompose the density matrix as a sum ofan unperturbed part and a perturbation which can be expanded in a series ofpowers of V (t):

ρ = ρ0 + 4ρ(t) (6.249)

ρ0 is the density matrix of the system in thermodynamic equilibrium at thegiven temperature. We sustitute now Eq. (6.249) in Eq. (6.247):

ih∂(ρ0 + 4ρ)

∂t= [ V, ρ0 + 4ρ ] = [ V, ρ0 ] + O(V 2) (6.250)

We assume that we only need to consider changes in the density matrix to firstorder. As a consequence, physical quantities of interest change only to thisorder.

It is customary at this point to introduce a mathematical procedure to ensureconvergence of the integrals over time involving the perturbation, known asadiabatic switching [26]: it is assumed that V s(−∞) = V s(+∞) = 0, ρs(−∞) =ρs(+∞) = ρ0, which can be obtained if V s(t) contains an exponential factore−ε|t| , ε = 0+.Let us integrate (6.250)

ih[ ρ(t) − ρ(−∞) ] =

∫ t

−∞

[ V (t′), ρ0 ]dt′ (6.251)

that is

ih∆ρ(t) =

∫ t

−∞

[ V (t′), ρ0 ]dt′ (6.252)

Let us now calculate the statistical average of the spin operator Sαl (in the

interation picture)

〈 Sαl (t) 〉 = Tr (Sα

l (t)ρ(t)) = Tr (Sαl (t)ρ0)

+Tr (Sαl (t)∆ρ(t)) (6.253)

6.14. DYNAMIC LINEAR RESPONSE 155

Since Tr (Sαl (t)ρ0) = Tr (Sα

l ρ0) = 〈 Sαl 〉0,

〈 Sαl (t) 〉 = 〈 Sα

l 〉0 + Tr (Sαl (t)∆ρ(t)) (6.254)

Traces are invariant under unitary transformations, so that all the traces abovecan be calculated in the SP where operators are time independent. We getfinally:

〈 ∆Sαl (t) 〉 = −γ

h

∫ ∞

−∞

e−ε|t′|θ(t− t′)dt′∑

β,m

Bβ0 (m, t′)G

(r)αβ(l,m; t− t′) (6.255)

The Green’s function appears in (6.255) because one can exploit the cyclic in-variance of the trace to obtain the commutator of the spin operators.


We define now the average local magnetization at time t as:

〈 M(Rl, t) 〉 =gµB

v0〈 Sl(t) 〉 (6.256)

where v0 = V/N is the atomic volume. Let us separate the induced part m inEq. (6.256):

m(Rl, t) = 〈 M(Rl, t) 〉 − 〈 M0(Rl) 〉 (6.257)

where〈 M0(Rl) 〉 = 〈 M0(Rl,−∞) 〉

is the equilibrium magnetization in the absence of the external field. Then:

mα(Rl, t) = −γ2

h

∑

l

∫ ∞

−∞

dt′e−ε|t′|∑

m

Bβ0 (m, t′)G

(r)αβ(l −m; t− t′) (6.258)

Let us Fourier transform Eq. (6.258) with respect to t and Rl :

mα(q, ω) =1

2π

∫

dt∑

l

e−i(q·Rl−ωt)m(l, t)

= − γ2

hv0

∑

l

∫

dte−iq·Rl

∫ ∞

−∞

e−ε|t′|dt′eiω(t−t′)eiωt′

×∑

h,β

Bβ0 (h, t′)G

(r)αβ(l −m, t− t′)

= −γ2

h

∫ ∞

−∞

dt′e−ε|t′|eiωt′

×∑

h,l,β

G(r)αβ(l − h, ω)Bβ

0 (h, t′)e−iq·Rl (6.259)


In equation (6.259) we have incorporated the adiabatic factor e−ε|t′| which“switches” on and off the perturbation V (t). We now substitute the inversespace Fourier transforms of G and B0 into (6.259):

Bβ0 (h, t′) =

1

N

∑

q

e−iq·RhBβ0 (Rh, t

′) (6.260)

and

G(r)αβ(l − h, ω) =

1

N

∑

q

e−iq·(Rl−Rh)G(r)αβ(q, ω) (6.261)

and we get

mα(q, ω) = − γ2

hv0

∑

K

G(r)αβ(q, ω)Bβ

0 (q + K, ω) (6.262)

where the sum over K = reciprocal lattice vectors, takes account of the fact thatq must belong to the first Brillouin zone, since m and G, being defined only onlattice sites, do not admit an expansion in arbitrarily small wavelengths. Thisrestriction of course does not apply to the external field, in principle, althoughin most cases the field wavelength will probably be much longer than a latticeconstant.


We define now the dynamic susceptibility tensor χ ′ as:

mα(q, ω) =

∫

d3q′χ ′αβ(q,q′;ω)Bβ

0 (q′, ω) (6.263)

So that, comparing (6.263) with (6.262), we have:

χ ′αβ(k,k′;ω) = χαβ(k;ω)

∑

K

δ3(k′ − k− K) (6.264)

where

χαβ(k;ω) = − γ2

hv0G

(r)αβ(k, ω) (6.265)

which is the main result of this section and, as anticipated, establishes theproportionality between the linear dynamic response function and the retardedGreen’s function.

6.15 Energy absorbed from external field

By definition, the power delivered by the radiation field to the spin system is

〈 Q(t) 〉 =d

dt(Trρ(H0 + V (t)) = Tr

dρ

dtH(t) + Trρ

dV

dt(6.266)

6.15. ENERGY ABSORBED FROM EXTERNAL FIELD 157


〈 Q(t) 〉 = −γ∑

l

∂B0(l, t)

∂t· 〈 Sl(t) 〉 (6.267)

Assume that either 〈 Sl 〉0 = 0 or 〈 Sl 〉0 ⊥ dB0/dt. Then,

〈 Q(t) 〉 = −γl∑ dB0(l, t)

dt·m(l, t) (6.268)

In terms of the Fourier tramsforms of B0 and m, we can calculate the totalenergy absorbed by the spin system over the whole infinite interval (−∞,∞),by assuming the adiabatic switching on and off of the external field. Then

∆Q = Q(∞) −Q(−∞) =

∫

〈 Q(t) 〉dt

= − γ

v0

∫

m(k, t)iωB0(−k,−ω)d3kdω (6.269)


We can express (6.269) in terms of the susceptibility:

mα(k, ω) =

∫

d3kχ ′(k, k′;ω)αβBβ0 (k′, ω) (6.270)

Then

∆Q = −i γv0

∫

d3k′∫

d3k

∫

dω ωχ ′(k, k′;ω)αβBβ0 (k′, ω)Bα

0 (−k,−ω) (6.271)

Since B0(−k,−ω) = B∗0(k, ω),

∆Q = i

∫

d3k

∫

d3k′[ ∫ 0

−∞

dω +

∫ ∞

0

dω

]

×ωBα∗0 (k, ω)χ ′

αβ(k, k′;ω)Bβ0 (k′, ω) (6.272)

Suppose that the external radiation field B0 has no large k components, andthat it is polarized along α. Then, q = q′ in equation (6.263) (or K = 0 in(6.264)). We can use Eq. (6.265) and the symmetry property of the Green’sfunction quoted in Eq. (6.37) to obtain:

∆Q =2

(2π)4

∫

d3k

∫ ∞

0

ωdω | Bα0 (k, ω) |2 Imχ(k, ω)αα (6.273)

This is a special case of the fluctuation-dissipation theorem.


6.16 Susceptibility of FM

As an example, let us obtain the transverse susceptibility of the Heisenberg FMin the RPA. Eq. (6.265) expresses the susceptibility tensor components in termsof retarded Green’s functions. In particular we have

χxx(k, ω) = −γ2ρ

hG+

xx(k, ω) (6.274)

where G+xx(k, ω) is the space-time Fourier transform of

G+xx(R, t) = 〈〈 Sx(R, t);Sx(0, 0) 〉〉+ (6.275)

We already found in Eq. (6.93) that

Gxx = 1/4(

G+− +G−+)

Upon use of Eq. (6.37) we find the relation

G−+(+)kω =

(

G+−(+)k,−ω

)∗

(6.276)

so that for Gxx we get the expression

Gxx(k, ω) = −σπ

Ek

(ω + iε)2 −E2k

(6.277)

where Ek = 2νJσ(1 − γk) + γB. In the absence of an external longitudinalfield (B = 0) we effectively find the soft mode associated with a global rotationof spins, a k = 0 magnon with vanishing excitation energy, so that the statictransverse susceptibility χxx ∝ k−2, as mentioned in Chap. 3 in connectionwith Goldstone’s theorem.

6.17 Corrections to RPA

The low temperature expansion of the magnetization as obtained from RPAyields for FM a spurious term in (T/Tc)

3, which is not present in the correctperturbation expansion obtained by Dyson [27]. Several attempts have beenmade to correct this flaw of the approximation. Callen’s alternative methodof decoupling [7] leads to agreement with the dominant terms of both Dyson’slow temperature expansion for the magnetization and Opechowski’s [28] high Texpansion for the magnetic susceptibility. For more details we refer the readerto the literature on this subject [3, 29].

References

1. Bogolyubov, N. N. and Parasyuk B. (1956) Dokl. Akad. Nauk. USSR109, 717.

6.17. CORRECTIONS TO RPA 159

2. Tyablikov, S. V., (1967) Methods in the Quantum Theory of Magnetism,Plenum Press, New York, Sect. 26.

3. Keffer, F. (1966) Handbuch der Physik, 2 Aufl. Bd. XVIII, Springer–Verlag, Berlin-Heidelberg.

4. Bogoliubov, N. N. and Tyablikov, S. V. (1959) Sov. Phys.- Doklady 4,604.

5. Hewson, A. C. and Ter-Haar, D. (1964), Physica 30, 890.

6. Tahir-Kheli, R. and Ter-Haar, D. (1962), Phys. Rev. 127, 38; 95.

7. Callen, H. (1963) Phys. Rev. 130, 890.

8. Yablonskiy, D. A. (1991) Phys. Rev B 44, 4467.

9. Abramowitz, Milton and Stegun, Irene (1965) Handbook of MathematicalFunctions, Dover Publications Inc., New York.

10. Gradshteyn, I. S. and Ryzhik, I. M. (1994) Table of Integrals, Series andProducts, Editor: Alan Jeffrey, Academic Press, San Diego.

11. Takahashi, M. (1990) Phys. Rev. B 42, 766.

12. Arovas, D.P. and Auerbach, A. (1988) Phys. Rev. B 38 , 316.

13. Bethe, H. A. (1931) Z. Phys. 71, 205.

14. Chakravarty, S., Halperin, B. I. and Nelson, D. R. (1987) Phys. Rev. B39, 2344.

15. Takahashi, M. (1989) Phys. Rev. B 40, 2494.

16. Kondo, J. and Yamaji, K. (1972) Prog. Theoret. Phys. (Japan) 47, 807.

17. L.Hulthen, L. (1938) Arkiv Mat. Astron. Fysik

18. Orbach, R. (1958) Phys. Rev. 112, 309.

19. Bonner, J. C. and Fisher, M. E. (1964) Phys. Rev. A 135, 640.

20. Wang, Y.-L. C. and Callen, H. B. (1964) J. Phys. Chem. Solids 25, 1459.

21. Devlin, John F. (1971) Phys. Rev. B4, 136.

22. Lines, M. E. (1967) Phys. Rev. 156, 534.

23. Narath, A. (1965) Phys. Rev. 140, A584.

24. Anderson, F. B. and Callen, H. B. (1964) Phys. Rev. 136, A1068.

25. Murau, T. and Matsubara, T. (1968) J. Phys. Soc. Japan 25, 352.


26. Kubo, R. (1957) J. Phys. Soc. Japan 12, 570.

27. Dyson, F. J. (1956) Phys. Rev. 102, 1217; idem, 1230.

28. Opechowski, W. (1959) Physica 25, 476.

29. Haas, C. W. and Jarrett, H. S. (1964) Phys. Rev. 135, A 1089.

Chapter 7

Dipole-Dipole Interactions

7.1 Dipolar Hamiltonian

We have been studying until now the exchange Hamiltonian

He = −∑

n6=m

JnmSn · Sm (7.1)

In dielectric magnetic materials, for which we expect (7.1) to be a good approx-imation, spins are well localized around each magnetic ion in the system. Theneach corresponding magnetic dipole is concentrated in a very small region ofspace, of a typical size smaller than any distance between the different dipolesin the system. We are then justified in using the point dipole approximationto describe the magnetic interactions. Usually the contribution to the magneticinteraction energy of higher order magnetic multipoles of the ions is negligiblein comparison with the dipolar one. The dipole-dipole interaction Hamiltonianis obtained by simply considering a magnetic dipole moment γSn = mn asso-ciated with each atom with spin Sn, and substituting it into the expression forthe mutual potential energy of a system of classical magnetic dipoles:

Hdip =γ2

2

∑

n6=m

1

R3nm

[

Sn · Sm − 3(Sn · Rnm)(Sm ·Rnm)

R2nm

]

(7.2)

where γ = gµB and Rnm = Rn − Rm. We can, on the other hand, consider(7.2) as a specific example of a bilinear form in the spin operators which obeysdefinite symmetry requirements. Consider the most general bilinear interation:

W =∑

l6=m

J αβlm Sα

l Sβm , (α, β) = (x, y, z) (7.3)

If the diadic J has cilindrical symmetry around the vector rl − rm ≡ Rlm itcan be written as:

Jαβ = Aαβ(| Rlm |)eαRα

lmeβRβlm

R2lm

(7.4)

161

162 CHAPTER 7. DIPOLE-DIPOLE INTERACTIONS

where eµ are the versors of thge coordinate axes and we have taken thecoefficient A symmetric under interchange of sites l and m. We can separate anisotropic part:

J = Aδαβeαeβ +BRlmRlm

R2lm

(7.5)

with A,B scalar functions of Rlm. If J is traceless,

3A+B = 0 (7.6)

Then:

J αβ = J(Rlm)eα(δαβ − 3RαlmR

βlm

R2lm

)eβ (7.7)

which has the form of the dipolar interaction. The traceless condition is equiv-alent to the statement that the dipolar potential satisfies Laplace’s differentialequation for Rlm 6= 0. The symmetry requirement imposed to get (7.7) is how-ever too restrictive for a lattice, where only some discrete rotations around Rlm

are symmetry operations. For example, for cubic crystals we might add to (7.6)a term [7]

beα(Rα

lm)2eα

R2lm

(7.8)

If the coeficient b depends on α, this term has the form of the anisotropicexchange already considered. The forms (7.7) and (7.8) are symmetric underexchange of l and m, but we could also construct quadratic forms which areanti-symmetric under permutation of two spins. For instance,

JA =∑

αβγ

eαεαβγDγ(lm)e

β (7.9)

is such a form if D is a pseudo-vector.The double contraction of (7.9) with spins Sl and Sm gives

Hm =∑

l6=m

Dlm · Sl ∧ Sm (7.10)

which is the Moriya-Dzialoshinsky interaction obtained in Sect. 5.2.Let us return to Eq. (7.2) and write Rnm in terms of circular components, so

that R±nm = xnm ± i ynm. Then the expression under the sum in (7.2) becomes

1

R3nm

(

S+n S

−m + S−

n S+m

2+ Sz

nSzm

)

− 3

R5nm

[

(1

2S+

n R−nm +

1

2S−

n R+nm + Sz

nznm )

× (1

2S+

mR−nm +

1

2S−

mR+nm + Sz

mznm)

]

7.1. DIPOLAR HAMILTONIAN 163

We expand the expression in square brackets above:

[· · ·] =1

4S+

n S+m(R−

nm)2

(7.11)

+1

4S−

n S−m(R+

nm)2 +1

4S+

n S−mR

−nmR

+nm

(7.12)

+1

4S−

n S+mR

−nmR

+nm +

(7.13)

+1

2S+

n R−nmznmS

zm +

1

2S+

mznmR−nmS

zn

(7.14)

+1

2S−

n R+nmznmS

zm +

1

2S−

mR+nmznmS

zn + Sz

nSzmz

2nm

Exercise 7.1Show that one can rewrite the dipolar Hamiltonian (7.2) as:

Hd =1

2γ2∑

n6=m

SznS

zm

R3nm

(

1 − 3z2nm

R2nm

)

− S+n S

−m

4R3nm

(

1 − 3z2nm

R2nm

)

− 3

4

(

S+n S

+mBnm + h.c.

)

− 3

2

(

S+n S

zmFnm + h.c.

)

(7.15)

The coefficients in Eq. (7.15) are defined as:

Bam =(R−

am)2

R5am

Fam =R−

amzam

R5am

(7.16)

Let us now obtain the RPA equations for the exchange-dipolar Hamiltonian,including the Zeeman interaction with an external field B. We must evaluate

the Green’s function G(r)a,b(t, t

′):

Grab(t, t

′) = −iθ(t− t′)〈[S+a (t);S−

b (t′)]〉 = 〈〈S+a ;S−

b 〉〉 (7.17)

As in Chap. 6 we obtain the equation of motion:

ihdGr

ab(t, t′)

dt= 〈〈[S+

a , H ];S−b 〉〉 + hδ(t− t′)〈[S+

a , S−b ]〉 (7.18)

Since H = He +Hd, we must now evaluate the commutator of S+a with Hd:


[S+a , Hd] = γ2

∑

m6=a

1

R3am

(1 − 3z2am

R2am

)[S+a , S

za ]Sz

m

− γ2

8

∑

m6=a

1

R3am

(1 − 3z2am

R2am

)[S+a , S

−a ]S+

m

− 3γ2

8

∑

m6=a

B∗am[S+

a , S−a ]S−

m − 3γ2

4

∑

m6=a

[S+a , S

za]S+

mFam

− 3γ2

4

∑

m6=a

[S+a , S

−a ]Sz

m + [S+a , S

za ]S−

mF ∗am (7.19)

The terms with coefficients F, F ∗ contribute higher order terms and we shallneglect them.To simplify the notation we drop the index r on the retarded Green’s functions.

Exercise 7.2Show that G+−

ab (ω) satisfies in the RPA, and with the same approximations asabove, the equation:

hωG+−ab (ω) = 〈Sz

a〉δab

π− 2

∑

m6=a

Jma

(

〈Sza〉G+−

mb (ω) − 〈Szm〉G+−

ab (ω))

+ γBG+−ab (ω) − γ2

∑

m6=a

(

1 − 3z2ma

R2ma

)

1

R3ma

〈Szm〉G+−

ab (ω)

− γ2

4

∑

m6=a

2

(

1 − 3z2ma

R2ma

)

1

R3ma

〈Sza〉G+−

mb (ω)

− 3γ2

42〈Sz

m〉∑

m6=a

B∗amG

−−mb (ω) (7.20)

whereG−−

ab (ω) = 〈〈S−a ;S−

b 〉〉(r) (7.21)

The appearance of this term reflects the fact that Hd does not conserve∑

m Szm.

The presence of G−− requires obtaining its equation of motion. It is moreconvenient to Fourier transform now all Green’s functions to k space. We write:

Gab(ω) =1

N

∑

k

Gk(ω)eik·(Ra−Rb) (7.22)

Exercise 7.3Obtain the system of two coupled equations for the Green’s functions G+−

k (ω),G−−

k (ω):

ωG+−k (ω) =

σ

π+ εkG

+−k (ω) + γBG+−

k (ω)

− [ γ2d0 +γ2

2dk ] σG+−

k (ω) − 3γ2

2σB∗

kG−−k (ω) (7.23)

7.1. DIPOLAR HAMILTONIAN 165

and

ωG−−k (ω) =

(

− εk + γB − γ2σ

(

d0 +γ2

2σdk

) )

G−−k (ω)

− 3γ2

2σBkG

+−k (ω) (7.24)

where

σ = 〈Szm〉

dk =∑

m6=a

1

R3am

(

1 − 3z2am

R2am

)

eik·Ram

Bk =∑

m6=a

(R−am)2

R5am

eik·Ram

εk = 2σ( J(0) − J(k) ) (7.25)

Calling

Rk = γB + εk − γ2σ

(

d0 +dk

2

)

(7.26)

we write the secular equation of the linear system (7.23) and (7.24) as:

[(E −Rk)(E +Rk)] +9

4γ4 | Bk |2 σ2 = 0 (7.27)

The roots of (7.27) are

ω2k = R2

k − 9

4γ4|Bk|2σ2 (7.28)

We shall see below that for the ferromagnetic, uniform, ground state to be stableagainst the excitation of these modes, we must require that:

Rk ≥ 0 , ∀k (7.29)

From the expression (7.28) for the dispersion relstion we see that we must have

| Rk |> 3

2γ2 | Bk | σ, ∀k (7.30)

for stability of the FM phase. If (7.29) and (7.30) are not satisfied we expectthe appearence of a different equilibrium phase. Cohen and Keffer [1] studiedthis problem within the FSWA and concluded that a s.c. FM lattice is unstable,while the FM b.c.c. and f.c.c. ones are at least metastable.The Fourier transforms of the dipolar tensor elements were calculated for aninfinite lattice. In this case, translation invariance makes all functions involvedindependent of the individual sites of each coupled pair. This simplification isclearly not possible if one or both sites are near an inhomogeneity like a surface


or in general any deffect which breaks that symmetry.We eliminate G−− in (7.23) and (7.24) to obtain G+−

k (ω), and find:

G+−k (E) =

σ

2πωk

E +Rk

E − ωk− E +Rk

E + ωk

(7.31)

One can now repeat the process of calculation we followed for the pure exchangeAFM and obtain the corresponding expressions for the zero-point spin deviationand Tc in the RPA. It is not possible however in this case to obtain analyticforms for Rk and Bk, except in the small k limit. Before evaluating R(k) andB(k) let us get a further physical insight of the exchange-dipolar magnons byappealing to the Holstein-Primakoff (HP) method of bosonization of the spinoperators which was applied in Chap. 4 to the pure exchange case.

7.2 Dipole-exchange spin-waves

Let us remind ourselves of the HP transformation equations:

S−i =

√2Sa†i fi

S+i =

√2Sfiai

Szi = S − ni (7.32)

and

fi =

√

1 − ni

2S

ni = a†iai .

The complete Hamiltonian is

H = He +HZ +Hd (7.33)

We express now the Zeeman, exchange and dipolar contributions to the Hamil-tonian in terms of the local boson operators. The Zeeman and exchange partswe have obtained already:

HZ = γBNS + γB∑

n

a†nan

He = −S2∑

l6=m

Jlm −∑

l6=m

Jlm

2S(a†l flfmam − a†l al) + a†lala†mam

(7.34)

Let us define the dipolar tensor for each pair of sites separated by x as

Dαβ(x) = − ∂2

∂xα∂xβ

1

| x | (7.35)

7.2. DIPOLE-EXCHANGE SPIN-WAVES 167

We write the dipolar part of H as

Hd =

3∑

i=1

H(i)d

where

H(1)d =

S2γ2

2

∑

R

′ Dzz(R) − γ2

4

∑

R

′ Dzz(Rlm) 2S( a†l flfmam

+ 2a†lal ) − 2a†lala†mam (7.36)

H(2)d = −3γ2

4

√2S∑

R

′F (R)flal(S − a†mam) + h.c. (7.37)

H(3)d = −3γ2S

4

√2S∑

R

′B(R)flalfmam + h.c. (7.38)

where a prime on a summation means that we exclude R = 0. The programnow consists of extracting from Eqs. (7.34) to (7.38) a quadratic form in theboson operators and then diagonalizing it to obtain the new normal modes.The quadratic form in the local basis which approximates to second order thecomplete Hamiltonian of Eq. (7.33) is

H2 = −2S∑

l6=m

Jlm(a†l am − a†lal) + γB∑

l

a†l al

− γ2

4

∑

l6=m

Dzz(lm)(a†lam + 2a†lal)

− 3γ2

4

∑

l6=m

( Blmalam + c.c. ) + C (7.39)

where the constant C is

C = −γBNS − S2∑

l6=m

Jlm

2+S2γ2

2

∑

l6=m

Dzz(lm) (7.40)

From H(2)d we get a linear term in the boson local operators but it vanishes

in an infinite crystal if the lattice is symmetric under one of the reflectionsx→ −x , y → −y or z → −z since the sum

∑

l6=m

Flm = 0 (7.41)

by symmetry in this case. We proceed now to diagonalize the quadratic Hamil-tonian (7.39). This is done in three stages:


1) Transformation to spin-waves.We go over to the Bloch (plane wave) representation

al =1√N

∑

k

e−ik·rlbk (7.42)

The transformed hamiltonian in k space is:

H2 =∑

k

Rkb†kbk +

1

2βkbkb−k

+1

2β∗

kb†kb

†−k (7.43)

where we simplified somehow the notation by calling

βk = −3

2γ2SBk (7.44)

2) Separate the k space into the subspaces kz > 0, kz < 0.

Then, we call∑(+) = summation over kz > 0.

Then

H2 = C +

(+)∑

k

Hk (7.45)

whereHk = Rk( b†kbk + b†−kb−k ) + βkbkb−k + β∗

kb†−kb

†k (7.46)

and Rk was defined in Eq. (7.26). We are explicitly using the symmetryRk = R−k, Bk = B−k .

3) Diagonalize Hk:Now we perform the linear Bogoliubov [2] transformation (section 3.3.2):

bk = ukck + vkc†−k

b−k = u−kc−k − v−kc†k (7.47)

We shall choose uk = u−k = real and

vk =| vk | eiαk (7.48)

One verifies that conmutators are preserved if

| uk |2 − | vk |2= 1 (7.49)

Substituting Eq. (7.47) in (7.46), one verifies that the new off-diagonal termsvanish if:

βku2k − 2Rkukv

∗k + β∗

k(v∗k)2 = 0 (7.50)

We shall show below thatβk =| βk | e−i2φk

7.2. DIPOLE-EXCHANGE SPIN-WAVES 169

where φk is the azimuthal angle of k in the coordinate system in which z istaken along the magnetization. If we substitute (7.48) into (7.50) and chooseαk = 2φk, we can rewrite (7.50) as a quadratic equation for the variable

z = |u/v| (7.51)

which has the solutions

z = Rk ±√

R2k − β2

k (7.52)

Exercise 7.4Show that (7.50) is compatible with (7.48) and (7.49) only if Rk > 0 , ∀k and| βk |<| Rk |.

This proves the assertion we made before Eq. (7.29).According to the condition (7.49) we can parametrize | u | and | v | as

| uk |= coshµk

| vk |= sinhµk (7.53)

To gain some insight into the physics of the excitations associated with theoperators ck, c

†k let us go back to the Hamiltonian in the form of Eq. (7.45).

The Heisenberg equations of motion of operators bk and b†−k are coupled. Whatwe are assuming implicitly upon making Bogoliubov transformation (7.47) isthat the linear combination of magnons (7.47) is a normal mode, which can

only happen if bk and b†−k have the same time dependence e−iEt. Substitutingexplicitly this time dependence we obtain for the eigen-modes the followingsystem of two equations:

(E −Rk)bk + β∗kb

†−k = 0

βkbk + (E +R(k))b†−k = 0 (7.54)


This system is identical with (7.23) and (7.24), if we disregard the inhomogene-ity on the right hand member of (7.23), so that we obtain the same eingenvalueequation (for low T ). From (7.54) we get for the eigenvalue

ωk =

√

R2k − 9

4γ2S2 | Bk |2 (7.55)

which coincides with Eq. (7.28) with the only difference that in RPA S is re-placed by the self-consistent average σ.


One verifies that Bogoliubov transformation restores the canonical harmonicform of the Hamiltonian:

H =

+∑

ωk ( c†kck + c†−kc−k ) (7.56)

Exercise 7.6Verify Eq. (7.56)

One finds with a little algebra that

vk = v−k =

√

Rk − ωk

2ωke−2iφk

uk = u−k =| βk |

√

2ωk(Rk − ωk)(7.57)

Let us quote a useful relation:

Rk ± βk cos 2φk

ωk= cosh2 µk + sinh2 µk

± sinhµk coshµk cos 2φk (7.58)

Exercise 7.7Verify Eq. (7.58)

We now proceed to calculate Rk and Bk and to evaluate the eigenmode en-ergies. This must be done in a special way for the uniform precession modewith k = 0, which is a magnetostatic mode. As we shall see shortly, the effect ofthe surface of the sample cannot be eliminated in this case, and the correspond-ing demagnetization field, due to the divergence of M at the surface, must beincorporated into the local field.

7.3 Uniform precession (k = 0) mode

Let us consider the sum:

d0 =∑

m6=0

(1 − 3z2m

R2m

)1

R3m

(7.59)

where we can choose the origin at the position of an arbitrary spin. This free-dom, as we already noticed, is due to the assumption that all spins are suffi-ciently far from the surface of the system. In fact, for a truly infinite crystalwith inversion symmetry, the sum in Eq. (7.59) is exactly zero, so that its con-tribution in a finite sample is entirely due to the surface. We use now Lorentzmethod to calculate d0. Let us define a separation parameter a as the radius ofthe Lorentz sphere. We require that the volume of the sphere be macroscopic,

7.3. UNIFORM PRECESSION (K = 0) MODE 171

that is a a0 = lattice constant. We divide the summation (7.59) into |R| ≤ aand |R| > a. For finite k we require aswell that ka 1, or λ a, where λ is themagnon wavelength. This means that within the Lorentz sphere we can considerthe magnetization as exactly (k = 0) or approximately (k 6= 0) uniform. Then,if the symmetry is cubic we shall neglect the contribution from points insidethe Lorentz sphere. We are then left with the sum (7.59) for |R| > a. Then| Rm |> a lattice constant, and we can substitute the sum by an integral:

d0 ≈ N

V

∫

|r|>a

d3r(1 − 3z2

r2)

1

r3=N

V

∫

|r|>a

d3r∇ ·( z

r3

)

(7.60)

with z ≡ zez and ez=versor of the z axis. Application of Gauss’ theorem yields:

d0 =N

V

∮

surface

z

r3· dS (7.61)

Figure 7.1: Boundary surfaces for the integration in Eq. (7.61).

Since the spherical region r < a is excluded, we have one internal and oneexternal boundary, the elements of area on each of them being oriented asindicated in Fig. 7.1. Then the surface integral splits into two parts:

d0 =N

V

∮

ext. bound.

z · dS

r3+N

V

∮

r=a

z · dS

r3(7.62)

The integral over the external boundary depends on the shape of the sample,and it is defined as

∮

ext. bound.

z · dS

r3≡ 4πNz (7.63)


which is the α component of the vector N defined as

4π(N)α =

∮

ext. bound.

(r)α(dS)α

r3(7.64)

It is easy to show that the sum rule

Nx +Ny +Nz = 1 (7.65)

follows from the fact that the solid angle subtended by a closed surface from aninterior point is 4π.

The coefficients Nα are called demagnetization factors.In an uniformly magnetized ferromagnetic sample which contains localized

magnetic moments mi, i = 1, · · ·N , we write the dipolar energy in the meanfield approximation as

Edip =1

2

N∑

i6=j

〈mλi 〉Dλµ(Rij 〈mµ

j 〉 (7.66)

Since〈mi 〉 = m , ∀ i

we have

Edip =1

2

N∑

i6=j

Dλµ(Rij)〈mλ 〉〈mµ 〉 (7.67)

We recognize the sum in Eq. (7.67) as the Fourier transform for k = 0 of thedipolar tensor. The eigenvalues of this tensor are

4π(N/V ) Nη

where η = X,Y, Z are the indices of the principal axes of the Fourier transformeddipolar tensor. Then the final form of Edip is:

Edip

V=

4π

2M2

(

NX cos2 α+NY cos2 β +NZ cos2 γ)

(7.68)

where cosα, etc., are the director cosines of the magnetization M referred tothe principal axes. One verifies that this expression can be written, in terms ofthe internal field defined in Eq. (7.67) for B = 0, as

Edip

V= −1

2Bi · M (7.69)

where the component of Bi along each principal axis λ = X,Y, Z is

BXi = −4πNX MX , etc.

Edip depends on the shape of the sample, as is clear from the definition of thedemagnetization factors, and it is called the “shape-anisotropy energy”.

7.3. UNIFORM PRECESSION (K = 0) MODE 173

Exercise 7.8Obtain (7.65) and show that the internal surface integral is −4π/3.

Then

d0 =N

V

(

4πNz −4π

3

)

(7.70)

Now we turn to Bk. The field inside the Lorentz sphere vanishes for cubicsymmetry. Then

Bk =N

V

∫

r>a

d3r(r−)2

r5eik·r

For k = 0 this simplifies if the sample has reflection symmetry (x → −x,y → −y, in which case

B0 ' N

V

∫

r>a

d3rx2 − y2

r5=

1

3

N

V

∫

r>a

d3r

[

(1 − 3x2

r2) − (1 − 3y2

r2)

]

1

r3=

4πN

3V(Nx −Ny) (7.71)

according to definition (7.64). The contributions from the surface integrals onthe Lorentz sphere cancel each other.On the other hand, for k = 0, (7.26) yields:

R0 = γB − 3

2γ2σd0 (7.72)

We call the frequency of the uniform precession mode ωu. Then from Eq. (7.28)we obtain

ω2u = (γB − 3

2γ2d0σ)2 − 9

4γ4σ2B2

0 (7.73)

We can now subtitute (7.71) and (7.72) into (7.73), and obtain

ωu =√

(γBi + 4πγNxM)(γBi + 4πγNyM) (7.74)

where the internal field Bi is defined as

Bi = B − 4πNz (7.75)

The magnetization M is

M = γσN

V(7.76)

While σ is replaced by S in the FSWA, we can calculate it self-consistentlywithin the RPA.

Exercise 7.9Obtain (7.75).


7.4 Eigenmodes for k 6= 0

In this case, we can expand the plane wave in both integrals of (7.25) in productsof spherical harmonics

eik·r =∑

l,|m|<l

−4π(+i)ljl(kr)Ym∗l (Ω)Y m

l (Ω′) (7.77)

where Ω ≡ (θ, φ) are the spherical angles of r and Ω′ ≡ (θk, φk) those of k , inthe same reference system in which the spin quantization axis z is parallel tothe magnetization. Substituting (7.77) into (7.25), and noticing that

1 − 3z2

r2≡ CY 0

2 (Ω)

we find:

dk = −CNV

∫

r>a

r2drjl(kr) ×∫

dΩ∑

lm

Y 02 (Ω)Y m∗

l (Ω)(i)l4π1

r3Y m

l (Ω′) (7.78)

Due to the orthonormality relations of the spherical harmonics

〈Y m∗l |Y m′

l′ 〉 = δll′δmm′

we obtain

dk = −4πC(i)2N

V

∫

r>a

drj2(kr)

rY 0

2 (Ω′) =

4πN

V(1 − 3cos2θk)

∫

r>a

drj2(kr)

r(7.79)

Now∫ x

drj2(kr)

r=j1(kx)

kx

In the casekR 1

where R is the size of the sample, j1(kR)/kR) is negligible at the surface,whatever the shape, and we drop the surface contribution. We remark that theresults that follow cannot be taken over to the case kR ≤ 1, which is the regionof the magnetostatic modes.We find

dk = 4πN

V(3 cos2 θk − 1)

j1(ka)

ka(7.80)

We notice that a was chosen with the condition ka 1, and

limx→0

(

j1(x)

x

)

=1

3

7.4. EIGENMODES FOR K 6= 0 175

so that

dk =4π

3

N

V(3 cos2 θk − 1) =

N

V

(

8π

3− 4π sin2 θk

)

(7.81)

Repeating the whole procedure for Bk we obtain

Bk = −NV

4π

3sin2 θke

−2iφk (7.82)

Upon substituting (7.65) and (7.70) into (7.28) we arrive at

ω2k = ( γB − 4πγMNz + εk ) ×

( γB − 4πγMNz + εk + 4πγM sin2 θk ) (7.83)

We conclude that the dominant contribution of dipolar interactions to themagnon energy is independent on | k | when ka 1 and kR 1. The last con-dition implies that the results obtained are valid away from the magnetostaticregion where kR ≤ 1. Notice that for any finite sample we cannot extend theseresults to the neighbourhood of k = 0.

We found a dispersion relation which depends on the orientation of k rela-tive to the magnetization. On the other hand, the exchange energy for small kis isotropic in a cubic crystal, and approximately quadratic. Then if we keepthe orientation of k fixed and vary the magnitude of k we shall obtain a curve,where the dependence on k comes through the exchange part. The series ofcurves thus obtained as we vary the orientation of k is called the magnon mani-fold. The interesting fact is that in many important cases the uniform precessionmode is degenerate with states in this manifold. There is then the possibilityof relaxing the uniform mode through processes in which a k = 0 magnon scat-ters against some static imperfection of the crystal and decays into one of thesedegenerate modes with k 6= 0. The extra linear momentum is transferred tothe static imperfection and the energy is conserved. This is the interpretationof experiments measuring the relaxation rate of the uniform mode. These arescattering events involving two magnons (two-magnon processes). In order thatthese processes conserve energy the uniform mode must be contained within themagnon manifold. Since the results we just obtained are shape dependent, weanalyze this for a spherical sample. In this case Eq. (7.74) yields the energy

ωu = γB.

This is positive for any external field, but one should keep in mind that thelocal field must point in the direction of the applied field if the system is to keepmagnetized in that direction, so that the minimum applied field which ensuressaturation of the sample is 4πM/3. Otherwise the sample will be subdividedinto magnetic domains, and the total net magnetization will be smaller thanthe saturation one at that temperature. Provided B > 4πMNz we verify thatif the sample is spherical ωu is contained within the magnon manifold bandfor the same B. The sphere is the most favourable shape of the sample forthis relaxation process. The degeneracy of the uniform mode with some of themodes within the manifold is verified for all ellipsoidal shapes of the sample,with the only exception of the disk nornal to the field.


7.4.1 Demagnetization factors

We list now the demagnetization factors and the values of ωu for various shapes.

• Sphere: Nx = Ny = Nz = 1/3 ; ωu = γB

• Infinite disk, B ⊥ disk : Nx = Ny = 0 , Nz = 1 ; ωu = γ(B − 4πM)

• Infinite disk, B ‖ disk: Nx = 1 , Ny = Nz = 0 ; ωu = γ√

B(B + 4πM)

• Infinite cylinder, B ‖ cylinder: Nx = Ny = 1/2 , Nz = 0 ; ωu =γ(B + 2πM)

• Infinite cylinder, B ⊥ cylinder: Nx = Nz = 1/2 , Ny = 0 ; ωu =

γ√

B(B − 2πM)

7.5 Ellipticity of spin precession

Let us look at the anisotropy of the spin precession around the quantization zaxis in a magnon mode. We must calculate the ratio of the variances of the xand y transverse spin components, that is the quantity

e =〈(Sx

i )2〉〈(Sy

i )2〉 = −〈(S+i + S−

i )2〉〈(S+

i − S−i )2〉 (7.84)

If we expand the local ascending and lowering operators in terms of spin waveoperators we have

S+i ± S−

i =

√

2S

N

∑

k

e−ik·Ri(b†−k ± bk) (7.85)

We want to evaluate the expectation values in Eq. (7.84) in a particular magnoneigenstate

˜| k〉 ≡ c†k˜| 0〉

where ˜| 0〉 is the ground state of the complete exchange-dipolar Hamiltonian.Substituting the Bogoliubov transformation from the bk to the ck one gets

ek =〈(Sx

i )2〉k〈(Sy

i )2〉k

=cosh 2µk + sinh 2µk − sinhµk coshµk cos 2φk

cosh 2µk + sinh 2µk + sinhµk coshµk cos 2φk

=Rk − βk cos 2φk

Rk + βk cos 2φk(7.86)

where use was made of Eq. (7.58). In a semi-classical image, the local spin is avector. During its precessional motion around the z axis the end point of this

7.6. EFFECT OF MAGNONS ON TOTAL SPIN 177

vector describes a periodic curve that we can project on the x, y plane. Theanisotropy of this curve is measured by the quantity e, or ek for a particularmagnon state as considered above. The expression (7.86) is maximum for φ =π/2 and minimum for φ = 0, so that the short axis of the curve is containedin the azimuthal plane containing k. We see that if the dipolar anisotropyparameter, defined as

Ed =γ2

Ja3(7.87)

is very small, so is the ellipticity

ε ≡| e− 1 | (7.88)

since in that case βk Rk.

7.6 Effect of magnons on total spin

In the analysis of relaxation processes it is important to distinguish betweenthose which change only the transverse or the longitudinal component of thetotal spin or both. Let us first calculate the effect of magnons on the square ofthe total spin vector operator,

S =

N∑

i

Si

In terms of the local boson HP operators the z component is

Sz = NS −N∑

i

a†iai (7.89)

and the transverse components can be approximated at low T by the lowest HPapproximation:

S+ ≈√

2S

N∑

i

ai =√

2SNb0 (7.90)

while the h. c. of this expression gives S−. For the z component

Sz = NS −∑

k

b†kbk (7.91)

if we make use of the canonical transformation to spin waves (7.42) in (7.90)and (7.91).

It will prove convenient to separate in (7.91) the term k = 0:

Sz = NS −∑

k 6=0

b†kbk − b†0b0 (7.92)


If we substitute in (7.91) the spin-wave operators bk in terms of the truemagnon operators ck we obtain for Sz the expression

Sz = NS −∑

k

a†iai = NS −∑

k

(| uk |2 + | vk |2)c†kck +∑

k

| vk |2) (7.93)

where the last sum is the zero-point spin-deviation due to the dipolar interaction.

We turn now to the other terms. If the temperature is not too high, or ingeneral if just a few magnons are excited in the system, the first term in (7.91)is much larger than the magnon contribution, so the average of the square isapproximately

〈(Sz)2〉 = (NS)2 − 2NS

∑

k 6=0

〈b†kbk〉 − 2NS〈b†0b0〉 (7.94)

while1

2〈S+S− + S−S+〉 = 2SN

(

〈 b†0b0〉 + 1/2)

(7.95)

Therefore

〈(S)2〉 = (NS)(NS + 1) − 2NS∑

k 6=0

〈b†kbk〉 (7.96)

We perform now the Bogoliubov transformation from the bk to the ck operators and obtain

〈(S)2〉 − (NS)(NS + 1) = −2NS∑

k 6=0

(| uk |2 + | vk |2)nk+ | vk |2 (7.97)

where nk = 〈c†kck〉 and the indicated average has been performed over an eigen-state of the exchange-dipolar Hamiltonian, or over the corresponding canonicalstatistical ensemble at temperature T . The last term above is the zero-pointcontribution to the spin-deviation, which describes the decrease of the magneti-zation of the ground state so this is a renormalization of S2 independent of theexcitation state. Therefore, the change of S2 from its ground state value due tothe presence of magnons is

∆S2 = −2NS∑

k 6=0

(| uk |2 + | vk |2)nk (7.98)

We see that the effect of a single magnon with k 6= 0 is to decrease the length ofthe total spin of the system, as well as decreasing the z component in one unit.This conclusion has important consequences regarding the physics of the differ-ent processes which are effective in relaxing the magnetization to its equilibriumvalue.

7.7. MAGNETOSTATIC MODES 179

7.7 Magnetostatic modes

We consider now the case in which the magnetization has a spatial variation ina scale comparable with the sample size R, that is kR ∼ 1. We are interestedin macroscopic samples and in the long-wave limit, so that we can still takeka 1, with a = lattice constant. Then the exchange terms in the energy,proportional to (ka)2, are negligible: we are still in the long wave limit asregards exchange. Let us also take kc ω, c = speed of light, so that wecan neglect the induced electric field and work in the magnetostatic limit ofMaxwell’s equations. In summary, we have a macroscopic ferromagnetic samplebelow Tc and we consider behaviour in the long wave limit of self sustainedmodes in the system which are the macroscopic (semi-classic) solutions of theconstitutive torque equation of Chap. 4 (with the adequate effective field actingon each spin) and Maxwell’s equations in the magnetostatic limit. We mustdetermine the solutions by imposing the boundary conditions of continuity ofthe normal component of the magnetic induction field and of the tangentialcomponent of the magnetic field at the surface. In this section we shall findconvenient to use the letter H for the magnetic field, as there is no confusionwith the Hamiltonian. The magnetostatic Maxwell’s equations are

∇∧ H = 0

∇ · (H + 4πM) = 0 (7.99)

and we also have the constitutive equation

dM

dt= γM ∧ H (7.100)

neglecting damping. Some authors use the negative sign in (7.100) but the choiceof sign will not have any consequence in the present context. The resultingmodes, called magnetostatic modes, were first experimentally found by Whiteand Solt [3] and Mercenau [4] in ferrite spheres and independently by Dillon[5] in ferrite disks, before they were described theoretically by Walker [6] in hiscalculations for samples with the shape of ellipsoids of revolution around the zaxis. These modes can be excited in a strongly non uniform oscillating magneticfield, which has a wavelength comparable to the sample size. The magnetostaticlimit allows us to take H = ∇ψ where ψ is a scalar potential. Besides, the systemis magnetized to saturation along z, so that the longitudinal component of Mis much larger that the transverse ones, and one writes

M = (mx,my,M0)

H = (hx, hy, H) (7.101)

The magnetic field must be substituted inside the sample by the effective field:

Heff = H0 +2A

M20

∇2M− 4πNzM (7.102)


The second term above, which is the exchange contribution to the effective field,was obtained in Chap. 4. We are adding the demagnetizing field as calculatedin the present chapter for the uniform magnetization case, since the longitudinalcomponent, which is assumed constant, is the dominant one.

We can now substitute (7.102) into (7.100), and linearize the constitutiveequation to find:

mk = χ(k)hk (7.103)

where the susceptibility tensor χ(k) is [7]

χ(k) =

χk −iKk 0iKk χk 00 0 0

(7.104)

and

χk =ωM

4πWk

Kk =ω

4πW 2k

ωM = 4πγM0

Wk = ωH + ωEa2k2

ωH = γ(H0 − 4πNzM0)

ωE =2Aγ

M0a2(7.105)

We are neglecting the anisotropy field and the damping terms. The eigenvaluesof the susceptibility tensor are χk ± Kk. For k ‖ M0 the eigenmodes are thetwo counter-rotating circularly polarized modes of the transverse magnetization.The difference between the two circular susceptibilities is responsible for theFaraday effect in the microwave region. The magnetic potential outside thesample satisfies Laplace’s equation. If one chooses a reference system with thex, y axes along two principal axes of the tensor χ(k) the potential inside satisfiesthe differential equation:

(1 + 4πχ0)

(

∂2

∂x2+

∂2

∂y2

)

ψ +∂2ψ

∂z2= 0 (7.106)

The boundary conditions are the continuity of ψ and of the normal componentof b = h + 4πm at the surface.

Walker [6] obtained the normal modes for an ellipsoid of revolution aroundz, which are indexed by three integers n,m, r. n and m are the indices of anassociated Legendre polinomial Pm

n in the expansion for ψ, while r numbers theroots of a secular equation.

References

1. Cohen, M. H. and Keffer, F. (1955) Phys. Rev. 99, 1135.

7.7. MAGNETOSTATIC MODES 181

2. Bogoliubov, N.N. (1947) J. Phys. USSR 9, 23; (1958) Nuovo Cimento 7,794.

3. White, R.L. and Solt, I.H. (1956) Phys. Rev. 104, 56.

4. Mercerau, J.E. (1956) Bull. Am. Phys. Soc. 1, 12.

5. Dillon, J.F. Jr (1956) Bull. Am. Phys. Soc. 1, 125.

6. Walker, L. (1957) Phys. Rev. 105, 390.

7. Keffer, F. (1966) Hand. der Phys., 2 Aufl. XVIII, 1.

Chapter 8

Coherent States of Magnons

8.1 Introduction

In Chap. 7 we studied states with a given number of magnons excited in aferromagnet with exchange and dipolar interactions. Therein we remarked thatthe expectation value of the transverse components of the local spin operatorvanishes:

〈nk | Sx,yi (t) | nk〉 = 0

This can be interpreted in the sense that the spins are not rotating coherently:it is as if in a state with a given number of magnons, which we shall herefromdenote as a “number state”, a random phase must be assigned to each spinwhich precesses around the magnetization axis with frequency ωk . Therefore,number states are not compatible with the semi-classical picture of a magnonas a classical wave as given in Sect. 4.4 . In the search for states that showa classical wave behaviour we must abandon the description in terms of stateswith a fixed number of magnons, and turn instead to coherent states.

The coherent states we shall presently define in analogy with those of photons[1] behave, in the limit of a large average number of magnons, as a classical spinwave.

We have seen in Chap. 4 that magnons behave like bosons, so that weshall first turn now to the definition and basic properties of a coherent stateconstructed as a linear combination of bosons.

8.2 Coherent states of bosons

We define a coherent state of bosons [2] as an eigenstate of the annihilationoperator:

ck | αk〉 = αk | αk〉 (8.1)

where we keep the notation of Chap. 7, since we shall be referring to exchange-dipolar magnons. Notice that we specialize to a particular k vector, because our

183

184 CHAPTER 8. COHERENT STATES OF MAGNONS

system has by assumption translation invariance, so Bloch’s theorem applies.The eigenvalue αk is a complex number.

We can expand the ket | αk〉 in the Fock basis of number states

| αk〉 =

∞∑

nk=0

Ank| nk〉 (8.2)

In Eq. (7.52) we expressed the magnetic Hamiltonian with exchange and dipolarinteractions as a superposition of independent harmonic oscillators, each of themcharacterized by a given k. This is true within the lowest order of the Holstein-Primakoff expansion (Sect 4.3). The boson annihilation operator satisfies therelation

ck | nk〉 =√nk | nk − 1〉 (8.3)

so from Eqs. (8.2) and (8.3) we get the following recursion relation for theexpansion coefficients:

αkAnk=

√nk + 1 Ank+1 (8.4)

If we assume that A0 6= 0 we can choose A0 = 1 as the initial condition forsolving the recurrence equation and we take care later of the normalization ofthe coherent state. The expansion then results

| αk〉 =∞∑

nk=0

αnk

k√nk

| nk〉 (8.5)

The state with nk magnons can be obtained upon operating repeatedly on thevacuum state with the creation operator, as:

| nk〉 =(c†k)nk

√nk!

| 0〉 (8.6)

Substituting (8.6) into (8.5) we get:

| αk〉 = eαk c†k | 0〉 (8.7)

Exercise 8.1Prove the following properties of the coherent state:

〈αk |= 〈0 | eα∗k ck (8.8)

〈αk | c†k = 〈αk | α∗k (8.9)

〈α′k | αk〉 = eα′∗

k αk (8.10)

c†k | αk〉 =∂

∂αk| αk〉 (8.11)

〈αk | ck =∂

∂α∗k

〈αk | (8.12)

8.2. COHERENT STATES OF BOSONS 185

From (8.10) we get the normalization constant for the coherent states. Thenormalized ones are defined accordingly as:

| αk〉 = e−12 |α|2 eαk c†

k | 0〉 (8.13)

There is no need to use a different notation for the normalized and the originalstates, since from now on we shall always use the latter. The overlap of twonormalized states is:

| 〈α | β〉 |2= e−|α−β|2 (8.14)

so although they are never exactly orthogonal, their overlap decreases exponen-tially as | α− β |→ ∞.


8.2.1 Overcompleteness of coherent states basis

In the following we shall drop the subindex k since we shall always deal withBloch states.

We shall prove the closure relation

A ≡∫

dα∗ dα

2πi| α〉〈α |≡ 1 (8.15)

First of all we verify that

[ c,A ] =[

A , c†]

= 0 (8.16)


All operators in Fock’s subspace of a given k vector are polynomials in ck , c†k,

and therefore also conmute with A, which accordingly satisfies the conditionsfor the application of Schur’s lemma [3], which states that: if a matrix (or anoperator on a linear space) A commutes with every matrix (or operator) of thespace, then it is a multiple of the identity:

A = a · 1 (8.17)

where a is some complex number. In order to determine a we calculate theexpectation value of A in the magnon vacuum Fock state:

a =

∫

dα∗ dα

2πi〈0 | α〉〈α | 0〉 (8.18)


Notice that for the normalized states 〈0 | α〉 = e−12 |α|2 so that we have:

a =

∫

dα∗ dα

2πie−|α|2 (8.19)

This is an integral over the whole complex plane of α. Instead of the variablesα and α∗ we may choose the real and imaginary parts of α = x + iy as theindependent variables. We find that a = 1, so that A = 1.

8.3 Magnon number distribution function

Let us now calculate the expectation value and the variance of the numberoperator n = c†c in a coherent state. For the expectation value we have:

n = 〈α | c† c | α〉 =| α |2 (8.20)

In order to calculate the mean of n2 we must first rewrite the product

n2 = c† c c† c

in normal order, that is, we must make the necessary permutations so that allcreation operators be on the left of the annihilation ones. The result is:

n2 ≡| α |2 + | α |4= n+ n2 (8.21)

The square of the variance is then

(∆n)2 = 〈α | (n− n)2 | α〉 =| α |2= n (8.22)

so that ∆n =| α | and∆n

n=

1√n

The internal product 〈n | α〉 is the probability amplitude for finding n magnonsin that coherent state:

〈n | α〉 =αn

√n!e−|α|2/2 (8.23)

and the probability distribution of n is the modulus squared of this quantity:

Pα(n) =| α |2n

n!e−|α|2 =

nn

n!e−n (8.24)

which is Poisson’s distribution function.

Exercise 8.4Verify that

∞∑

n=0

Pα(n) = 1 (8.25)

8.4. UNCERTAINTY RELATIONS 187

8.4 Uncertainty relations

We show now that the coherent state is the equivalent, for a harmonic oscillator,of the minimum wave packet for a free particle: the product of the variancesof the pair of canonically conjugated operators c, c† in a coherent state hasthe minimum value allowed by Heisenberg uncertainty principle. It is veryconvenient at this point to introduce the pair of so called “quadrature” operatorsdefined as:

X =1

2(c† + c)

Y =i

2(c† − c) (8.26)

with commutator

[X, Y ] =i

2(8.27)

so that the quadratic boson Hamiltonian

Hq = hωq(nq +1

2) (8.28)

can be written as

Hq = hωq(X2 + Y 2) (8.29)

In a number state, we have:

〈n | X2 + Y 2 | n〉 = (n+1

2) (8.30)

while

〈n | X | n〉 = 〈n | Y | n〉 = 0 (8.31)

The variance of an arbitrary operator A in a given state | s〉 is

(∆A)2 ≡ 〈s | (A− 〈s | A | s〉)2 | s〉 (8.32)

Then we find that in the number state

(∆X)2 = (∆Y )2 =1

2(n+

1

2) (8.33)


The form of the uncertainty relation for the quadrature operators in a num-ber state is therefore:

| ∆X | | ∆Y |= 1

2(n+

1

2) ≥ 1

4(8.34)


On the other hand:

Exercise 8.6Show that the expectation values of the quadrature operators calculated in a nor-malized coherent state are:

〈α | X | α〉 =| α | cos θ

〈α | Y | α〉 =| α | sin θ (8.35)

where θ is the phase of the complex number α.

We can obtain now the variances in the coherent state:

Exercise 8.7Show that:

〈α | ( X − 〈α | X | α〉 )2 | α〉 = 〈α | ( Y − 〈α | Y | α〉 )2 | α〉 =1

4(8.36)

This result implies that in the coherent state the product of the variances ofX and Y has the minimum value allowed by the uncertainty principle. To seethis, let us remind that given a pair of non-commuting operators A,B such that[A,B] = K = number, the product of the variances calculated in a given statesatisfies:[4]

∆A∆B ≥ 1

2| 〈K〉 | (8.37)

The equality is obtained if the states chosen to calculate the expectation valueshave that special property for the pair of operators considered. Such states,are the analog of the ”minimum uncertainty wave packet” for a free particle.Eq. (8.36) shows that this is the case for the quadrature operators in any co-herent state, independently of the value of α. For a number state this propertyonly obtains, in the vacuum (n = 0) case.

8.5 Phase states

We shall follow the theory of the phase states of the electromagnetic field [2]and define the phase operator through the separation of amplitude and phase ofthe annihilation operator:

ck = (nk + 1)1/2eiφ (8.38)

which is equivalent to

eiφ ≡ (nk + 1)−1/2ck (8.39)

Assume φ† = φ. Then, upon taking the hermitian conjugate of Eq. (8.39) wehave:

e−iφ = c†k(nk + 1)−1/2 (8.40)

8.5. PHASE STATES 189

and we also find thateiφe−iφ = 1 (8.41)

although we cannot invert the order of the product.

Exercise 8.8Prove Eq. (8.41)

One verifies the following properties:

eiφ | n〉 =| n− 1〉 , n > 0 ;

eiφ | 0〉 = 0

e−iφ | n〉 =| n+ 1〉〈n | eiφ | n+ 1〉 = 〈n+ 1 | e−iφ | n〉 = 1 (8.42)

We remark that〈n | eiφ | n+ 1〉 6= ( 〈n+ 1 | eiφ | n〉 )∗ (8.43)

so that the exponential operator is not Hermitian. Instead,

Exercise 8.9Prove that cos φ and sin φ are Hermitian, and that

〈n− 1 | cos φ | n〉 =1

2= 〈n | cos φ | n− 1〉

〈n− 1 | sin φ | n〉 =1

2i= −〈n | sin φ | n− 1〉 (8.44)

Therefore these are observables, although they cannot be measured simultane-ously as shown by their non-vanishing commutator:


[ cos φ , sin φ ] =1

2i

(

c†(n+ 1 )−1 c− 1)

(8.45)

and verify that all matrix elements of the commutator vanish, except for thediagonal ground-state matrix element, which is:

〈0 | [ cos φ , sin φ ] | 0〉 = − 1

2i(8.46)

Other useful commutators are:

[ n , eiφ ] = −eiφ

[ n , e−iφ ] = e−iφ

[ n , cos φ ] = −i sin φ


[ n , sin φ ] = cos φ

Since the number and the phase operators don’t commute the amplitude andthe phase of the transverse components of the local spin operator cannot besimultaneously precisely determined. The corresponding uncertainty relationscan be easily obtained from the definition of the variances, following the usualprocedure based on Schwartz’s inequality[4]:

∆n∆cosφ ≥ 1

2| 〈sinφ〉 |

(8.47)

∆n∆sinφ ≥ 1

2| 〈cosφ〉 | (8.48)

8.6 Magnon states of well defined phase

We have seen that cos φ and sin φ do not commute. However, there is onlyone non vanishing matrix element of their commutator in the number basis.This justifies the search for states which are approximately eigenstates of bothoperators.

We define

| φs〉 =1√s+ 1

s∑

n=0

einφ | n〉 (8.49)

and

| φ〉 = lims→∞

| φs〉 (8.50)


lims→∞

〈φs | φs〉 = 1 (8.51)

In the limit defined above, we find that

〈φs | cos φ | φs〉 → cosφ+ O(s−1)

〈φs | sin φ | φs〉 → sinφ+ O(s−1) (8.52)

One also finds that to order s−1/2,

〈φs | cos2 φ | φs〉 → cos2 φ

〈φs | sin2 φ | φs〉 → sin2 φ (8.53)

From these results we conclude that the state | φs〉 behaves approximately as

a simultaneous eigenstate of cos φ and sin φ. Besides, as a consequence of Eqs.(8.52) and (8.53), it turns out that the uncertainties of both operators in thephase state vanish in the limit s→ ∞.

8.7. PROPERTIES OF THE SINGLE-MODE NUMBER STATES 191

8.7 Properties of the single-mode number states

Let us consider a single-mode (which implies a specific k ) excited in a FM, witha given number n of magnons. Clearly, for such a state the uncertainty in thenumber, ∆n is zero. For the phase operators we find:[2]

〈n | cos φ | n〉 = 〈n | sin φ | n〉 = 0 (8.54)

while

〈n | cos2 φ | n〉 = 〈n | sin2 φ | n〉 =1

2for n 6= 0

=1

4for n = 0. (8.55)

We find that, leaving aside the n = 0 state, the uncertainties are

∆ cos φ = ∆ sin φ =1√2

(8.56)

which implies that the number state phase is completely undetermined, and canhave any value between 0 and 2π.

These results lead to an immediate physical interpretation of the propertiesof the transverse components of the local spin operator.

Returning to Eq. (4.22),

S+(r) =

√

2S

N

∑

k

eik.rbk (8.57)

where higher order terms of the Hosltein-Primakoff expansion have been ne-glected, we express bk in terms of the exchange-dipole magnon operators ck, c

†k,

according to Eq. (7.43). We transform back from polar to cartesian componentsof S and finally rearrange the terms in the k summation, to get:

Sx(r) =

√

S

2N

∑

k

(uk − v∗k)(

eik.rck + e−ik.rc†k

)

(8.58)

The expectation value of S2x(r) in the single-mode number state | nk〉 is:

〈nk | S2x(r) | nk〉 =

S

N(nk +

1

2) (uk − v∗k)2 (8.59)

The amplitude of the transverse components is therefore proportional to n1/2

for large n.On the other hand, we have just seen that the phase is completely undefined

for this state.


8.8 Properties of a single-mode phase state

We have seen that the uncertainties of the cos φ and sin φ operators vanish in aphase state.

Let us now obtain the variance of the number operator. First we calculate:

lims→∞

〈φs | n | φs〉 = lims→∞

s∑

n1

s∑

n2

〈n1 | n | n2〉ei(n1−n2)φ =

= lims→∞

1

s+ 1

s∑

n

n = s/2 (8.60)

Let us now calculate the average of n2:

lims→∞

〈φs | n2 | φs =s(2s+ 1)

6(8.61)

Finally,

(∆n)2 = 〈n2〉 − (〈n〉)2 =s2

12(8.62)

For the r. m. s. relative deviation we find

∆n

〈n〉 =

√

1

3(8.63)

As to the expectation value of the local spin components, we find, before takingthe limit s→ ∞:

〈φs | Sx(r) | φs〉 =2

3(s+ 1)1/2 (8.64)


The same result obtains for Sy(r).In conclusion, we have shown that in a single-mode phase state the trans-

verse components of magnetization have a well defined phase but a divergentamplitude.

8.9 Expectation value of local spin operators ina coherent state

We shall change now to the Heisenberg representation for the transverse localspin operators, in order to display the traveling wave characteristic of these exci-tations. In this representation, the magnon creation and annihilation operatorsat time t relate to those at time t = 0 as:

ck(t) = ck · e−iωkt

c†k(t) = c†k · e−iωkt (8.65)

8.9. EXPECTATION VALUE OF LOCAL SPIN OPERATORS 193

With the use of Eqs. (4.22) (and its hermitian conjugate) and (7.43) we

can express the transverse operators Sx,y(r, t) in terms of ck(t), c†k(t) (noticethat the operators ak in (4.22) are called bk in (6.43) , and then calculate theirexpectation values in a coherent state | α〉, to obtain [5]:

〈αk | Sx(r, t) | αk〉 = Sk cos (r · k − ωkt+ θk)

〈αk | Sy(r, t) | α〉 = ekSk sin (r · k − ωkt+ θk) (8.66)

where θk is the phase angle of αk, and we have defined:

Sk =

√

2S

N(uk − vk) | αk |

ek =uk + vk

uk − vk=

(

Rk+ | βk |Rk− | βk |

)1/2

(8.67)

We are assuming for simplicity that uk and vk are real.


In conclusion, we have shown that in a coherent state with wave vector kthe transverse components of the spin operator behave as those of a trans-verse travelling wave with the frequency and the ellipticity already found inChap.7. The physical possibility of experimentally exciting macroscopic coher-ent states was discussed in Ref. [5]. These states have since been observed inseveral experiments.[6]

References

1. Glauber, R. J. (1963) Phys. Rev. 131,2766.

2. Loudon, Rodney (1978) “The Quantum Theory of Light”, ClarendonPress, Oxford.

3. Heine, Volker (1977) “Group Theory in Quantum Mechanics”, PergamonPress.

4. Messiah, Albert (1959) “Mecanique Quantique” v.1, pg. 379, Dunod,Paris.

5. Zagury, Nicim and Rezende, Sergio M. (1971) Phys. Rev. B4, 201

6. Tsoi, M. et.al. (2000) Nature 406, 46; Melnikov, A. et.al. (2003) Phys.Rev. Lett. 91 , 227403; Zafar Iqbal M. et.al. (1979) Phys. Rev B20,4759; Weber, M. C.et.al. (2006) J. App. Phys 99, 08J308.

Chapter 9

Itinerant Magnetism

9.1 Introduction

We have considered up to now insulating (or dielectric) systems, where electronspins are well localized. The models we used do not describe therefore systemswhere all or at least a finite fraction of the electrons are itinerant, as in metals.And yet the most common ferromagnetic materials are metallic, namely Fe,Niand Co. In this chapter we shall present some of the simplest models for themagnetism of metals. Those metals which display ordered magnetic phases donot in fact lend themselves to a description based exclusively upon extendedelectron states. As I shall mention later on, realistic models for these systemsrequire a formulation capable of representing their dual extended and localizedcharacter.

I shall first describe briefly the simpler problem of the paramagnetism of ametal with a non-degenerate (s state) conduction band.

9.2 Pauli paramagnetic susceptibility

We consider for the time being independent electrons in a single band and derivein what follows the expression for their static, zero field, magnetic susceptibility.[1] In this case the electron states are completely specified by their wave-vectorand spin. Under an external applied magnetic field B an electron with wavevector k has a spin dependent energy:

εkσ = εk − µBσB (9.1)

and the spin-up and spin-down bands are now relatively shifted by a constantamount.

Let N(E) be the total average number of electrons with energy ≤ E, thatis, the sum of the averages N± for both spins. In the presence of B we have:

N± =1

2N(EF ± µBB) (9.2)

195

196 CHAPTER 9. ITINERANT MAGNETISM

We verify that this is consistent with the fact that for zero field the numbers ofup and down spin electrons are equal. We shall neglect the dependence of EF

on B, since we shall limit ourselves to effects which are of first order in B. Sincethe Zeemann term of the energy is small compared with EF we can expand ther. h. s. of Eq.(9.2) in powers of the Zeemann term, and keep only the first:

N± ≈ 1

2N(EF ) ± µBB

(

dN

dE

)

EF

(9.3)

For a degenerate metal all states of spin σ are occuppied up to the chemicalpotential

µσ = EF + σµBB

so that the average number of electrons of that spin with energy less or equalto some value E is equal to the number of available states up to that energy,which we call Z(E), which implies

N± =1

2Z(EF ) ±

(

dZ

dE

)

EF

(9.4)

If Z(E) is refferred to unit volume, the magnetization is

M = µB(N+ −N−) = 2µ2BB ρ(EF ) (9.5)

where ρ(EF ) =(

d Zd E

)

EF. We find now that Pauli susceptibility is [1]

χP = 2µ2B ρ(EF ) (9.6)

9.3 Stoner model of ferromagnetic metals

In order to deal with itinerant electron states it is convenient to consider con-tinuous space distributions of charge and spin. In particular, if we want tocalculate the total spin ∆S contributed by electrons in a volume ∆V around apoint R in space, we can integrate the spin volume-density operator over ∆V :

∆S =

∫

∆V

d3x ~σ(R + x) (9.7)

where the spin density operator for a system with a total number N of electronsis

~σ(r) = h

N∑

i=1

S(r − xi) (9.8)

and we define the operator

S(r − xi) = δ3(r − xi)~σi (9.9)

xi are electron coordinate operators and σαi are Pauli matrices (α = +,−, z).

According to the rules for second quantizing the one-particle operator S(r − xi)

9.3. STONER MODEL OF FERROMAGNETIC METALS 197

[3] , we represent ~σ as a series of terms, each of which is the product of its matrixelement 〈s | ~σ | t〉 between one particle states s, t in any complete orthonormalset (c. o. s.) which spans the one particle Hilbert space, times the product a†sat

of the second quantized operators which destroy the initial state t and createthe final one s:

~σ(x) =∑

ν,α;µ,β

a†µ,αaν,β ~σµ,α;ν,β(x) (9.10)

where the matrix element is

~σµ,α;ν,β(x) =

∫

d3xelφ∗µ(xel)〈α | hδ3(r − xel)~σφν(xel) | β〉

= hφ∗µ(x)~σαβφν(x) (9.11)

and ~σαβ are the αβ matrix elements of the three Pauli matrices. In the sum(9.10) α, β = ±1 corresponding to the two eigenstates of the z componentof the spinor. The basis states in Eq. (9.10) are spin-orbitals, products of awave function φ and a spinor | α〉 as explicitly indicated in Eq. (9.11). For atraslationally invariant system it is convenient to use the Bloch basis [4]

φk,ν,α = eik·x uk,ν(x) | α〉 (9.12)

in which case

~σ(x) =∑

k,q

∑

µν

∑

αβ

e−iq·x u∗k+q,µ(x)~σα,βuk,ν(x) a†k+q,µαak,νβ (9.13)

The u factor above is a periodic function of its argument, and the secondsubindex corresponds to the various (in principle infinite) bands. The sum-mation over k and q above extends over the volume of the first Brillouin zone(BZ). If we use plane waves instead of Bloch waves the factors u are replaced bya constant, determined simply by the normalization condition chosen, while thek space is extended up to infinity. The crystalline structure can be taken intoaccount in this case by imposing that all wave vectors which are not within thefirst BZ are folded back into it by subtracting the adequate reciprocal vector G(empty lattice model). This procedure yields a unique result for each arbitraryk. In the plane wave representation, we can write

~σ(x) =∑

q

eiq·x~σ(q) (9.14)

whereσα(q) =

∑

p s t

a†p+q, sσαstap t (9.15)

and s, t are spin indices. The circular components of σ are then

σ+(q) =∑

p

a†p+q↑ap↓


σ−(q) =∑

p

a†p+q↓ap↑

σz(q) =1

2

∑

p

(

a†p+q↑ap ↑ − a†p+q↓ap↓

)

(9.16)

9.4 Hubbard Hamiltonian

In order to proceed, we must now define the Hamiltonian for the metal. Forthe case of the 3d transition metals, Hubbard [5] proposed a simplified formof the electron-electron interaction potential by discarding, as relatively lessimportant, the electrostatic Coulomb repulsion among electrons localized ondifferent sites, which is expected to be reasonably screened by the less localizeds and p electrons. The orbital degeneracy of the d orbitals demands in principlea representation with several states on each site, but as a first simple approachone works with the single orbital case chosen above. The remaining terms inthe Hamiltonian are one electron terms, including the atomic binding energy ofone electron and the hopping to the neighbouring ions due to the self-consistentaverage one-electron potential. In the site representation the Coulomb on-siterepulsion is parametrized by a constant U , which is usually fitted by comparisonwith experiment. The simplest (one band) Hubbard Hamiltonian in the Wannierrepresentation is then [5]

H = E0

∑

iσ

a†iσaiσ +∑

i,jσ

tij

(

a†iσa†jσ + c.c.

)

+U∑

i

ni↑ni↓ (9.17)

where E0 and the hopping terms tij are adequate band parameters, which in thetight-binding approximation are taken from atomic parameters. In particularthe diagonal term can be approximated by the atomic ground state energy.Changing to the Bloch representation through the unitary transformation

ai =1√N

∑

k

ake−ik·Ri

ak =1√N

∑

i

aie+ik·Ri (9.18)

we have

H =∑

kσ

εka†kσ +

U

N

∑

pkq

a†p+q↑ap↑a†k−q↓ak↓ (9.19)

where

εp = E0 +∑

Rij

tij eiRij·p (9.20)

9.5. INSTABILITY OF PARAMAGNETIC PHASE 199

In the Hartree–Fock approximation, the electrons are independent particleswhich occupy states with energies

Epσ = εp +U

N

∑

k

fk,−σ (9.21)

The second term on the right hand side of (9.21) is the average self-consistentfield acting on each electron due to all the other ones in the metal, withinthe on-site truncated Hubbard model. The energies Epσ are the quasi particleenergies in the Hartree–Fock approximation to Hubbard Hamiltonian. In (9.21)

fkσ ≡ 〈a†kσakσ〉 is the Fermi distribution

fkσ =1

eβ(Ekσ−µ) + 1

for quasiparticles with energy Ekσ and spin σ. To impose self-consistency wedemand that the Fermi level µ be determined by the condition that the totalaverage number of particles has a given value, assumed known, n = Nel/N . Weshall calculate the average number of electrons with spin σ per atom:

1

N

∑

k

fkσ =1

N

∑

k

nkσ = nσ (9.22)

N being the total number of atoms in the system. Then we must have

n = n↑ + n↓ (9.23)

In the paramagnetic phase, n↑ = n↓. We may ask ourselves whether this phaseis stable with respect to an infinitesimal ferromagnetic breaking of spin-up ↔spin-down symmetry [6]. The Hamiltonian (9.19) was chosen to represent thecompetition between the tendency of the kinetic energy term to de-localize elec-trons and the opposite effect of the intra-atomic repulsion U . The latter favoursstates with just one electron, or none, on each atom. Hopping of electrons be-tween different ions clearly requires some double occupancy, so the two termshave opposite effects. In the 3d metals the d band width is W ≈ 4 eV , whileU ≈ 1–3 eV , so U/W < 1. In the 4f metals the f bandwidth is very small,while U ≈ 5–6 eV so that U/W > 20. In the 5f metals, W decreases from2.5 eV to 0.5 eV on going from Ac to Bk while U varies between 2 eV and10 eV [8] so that U/W is between 1 and 7. As we see, the application to allthese metals of Hubbard model requires its solution not only in the two oppositelimits of weak and strong correlations, but also in the intermediate case. Theexact solution of Hubbard’s Hamiltonian is unknown, except for the 1d case [9].In the rest of this chapter we shall develop the Hartree–Fock solution of (9.19),which is strictly valid for the weak correlation case U/W < 1.

9.5 Instability of paramagnetic phase

Let us imagine that we excite the down spin electrons which have an energycontained within a shell of thickness δE below the Fermi level µ and replace


them in the unoccuppied states of the up spin band, just above µ. Let usnow calculate the energy change per atom due to this operation, which can bethought as an infinitesimal splitting of the bands. The change of kinetic energyper atom is

∆Ekin = ρ(µ)(δE)2 (9.24)

where ρ(E) is the density of electron states of energy E per atom. The variationof the interaction energy, within the Hartree–Fock approximation, is

∆Eint/U = (n/2 + ρδE)(n/2 − ρδE) − n2/4 =

−ρ2(µ)(δE)2 (9.25)

The total change in energy per atom is

∆E = ρ(µ)(δE)2 (1− Uρ(µ)) (9.26)

The condition for the instability of the non-magnetic state is then [7]

Uρ(µ) ≥ 1 (9.27)

It is interesting to remark that Eq. (9.26) is the same for either up or down spin,reflecting the time reversal symmetry of the model. Condition (9.27) is calledStoner condition. The main concepts of this model of itinerant ferromagnetismwere developed by Slater [10] and Stoner [11] around 1940, and the resultingtheory is known as Stoner’s model of FM metals. In the next section we shallcalculate the dynamic transverse susceptibility χ−+(ω,q) in the paramagneticstate and we shall find that the static uniform susceptibility in the non-magneticlimit diverges for Uρ(µ) = 1.

Within this model, the energy band, which in the paramagnetic state isthe same for both spins, is split into a pair of bands, each containing statesof a given spin, which are identical to the original paramagnetic bands, onlymutually displaced, so that the lower band contains more electrons, becomingthe majority band, thereby creating a state with a finite magnetization. Onecan distinguish between two cases of split band FM. We define the parameter

∆0 = U(n↓ − n↑) ≡ Um (9.28)

where m = 〈Sz〉 is the average of the z component of spin per atom and weare assuming that the sample magnetization points down, so that m > 0. ∆0

is called the exchange splitting . If ∆0 > µ, the minority spin band will becompletely empty at low T . When this happens, one speaks of (a) completeferromagnetism , as compared to (b) partial or incomplete ferromagnetism inthe opposite case. Both cases are schematically shown in Figs. 9.1 and 9.2.

9.6 Magnons in the Stoner model

9.6.1 The RPA susceptibility

Let us suppose that the system has a ferromagnetic ground state. We assumethat the temperature is well below Curie’s Tc, so that a well defined magne-

9.6. MAGNONS IN THE STONER MODEL 201

Figure 9.1: Split-band incomplete ferromagnetic state.

tization axis exists, and we shall consider a one domain situation, that is, theaverage magnetization is constant throughout the sample. We shall obtain adescription of a magnon in an itinerant ferromagnetic system as a travellingwave of transverse fluctuations of the local magnetization.

Let us assume that in the ground state the majority spins point down. Nowconsider the retarded spin-flip propagator (see Chap. 6) [13]:

χ−+(q, t) = iθ(t)〈[

σ−(q, t), σ+(x = 0, t = 0)]

〉=

∑

k

χ−+(k,q, t) (9.29)

where we define

χ−+(k,q, t) = 〈〈a†k+q↓ak↑ ;σ+(~x = 0, t = 0) 〉〉 (9.30)

with the notation of Chap. 6. We limit ourselves from now on, for simplicity,to the one-band case and we eliminate accordingly the band index from theoperators.

The equation of motion for the propagator defined in (9.29) is

ih∂

∂tχ−+(k,q, t) = −δ(t)〈

[

a†k+q↓ak↑ , σ+(x = 0, t = 0)

]

〉

+ 〈〈[

a†k+q↓ak↑ , H]

;σ+(~x = 0, t = 0)〉〉 (9.31)


Figure 9.2: Split-band complete ferromagnetic state.

where H is Hubbard’s Hamiltonian (9.19). Upon substituting H in the commu-tator of Eq. (9.31) we obtain several terms with four electron operators.

As in the case of the RPA for the Heisenberg Hamiltonian we truncate thecorresponding hierarchy of Green’s-function integro-differential equations by ap-plying the RPA appropriate for this case. This consists of reducing all fourelectron operator terms to products of two operators, by substituting every pairof one creation and one annihilation operator by its statistical average. In orderto do this factorization consistently one selects all possible different products ofone creation and one annihilation operator from each four operator term in allpossible ways, and adds all the terms obtained with the sign corresponding torespecting the anticommutation rules of fermionic operators. This is easily doneby keeping track of the parity of the number of permutations needed to bringboth operators selected to adjacent positions in the product, and assigning tothe corresponding term in the sum the sign + for even, − for odd permutations.As a result, Eq. (9.31) becomes:

[

ih∂

∂t+ (Ep+q↑ −Ep↓)

]

χ−+(p,q, t)

= −δ(t) [fp+q↑ − fp↓] − (fp+q↑ − fp↓)U

N

∑

k

χ−+(k,q, t) (9.32)


The one-electron energiesEk,σ appearing in (9.32) are those defined in (9.21).We Fourier transform (9.32) with respect to the time variable, obtaining

χ−+(p,q ;ω) =(fp↑ − fp+q↓) (1 + Uχ−+(q, ω))

ω +Ep+q↑ −Ep↓(9.33)

where we introduced the quantity

χ−+(q, ω) =1

N

∑

p

χ−+(p,q, ω) (9.34)


Equation (9.33) is an inhomogeneous linear integral equation with a sep-arable kernel, which can be easily solved. The quantity we want is χ−+(q, ω)defined in (9.34, and we find:

χ−+(q, ω) =χ−+

0 (q, ω)

1 − Uχ−+0 (q, ω)

(9.35)

where

χ−+0 (q, ω) =

1

N

∑

p

fp+q↑ − fp↓ω − ∆p,q

(9.36)

and

∆p,q = Ep+q↑ −Ep↓ (9.37)


We have now obtained the Fourier transform of the spin propagator definedin (9.29), within the RPA. Since we want the retarded propagator, we add asmall positive imaginary part to ω. If U → 0, (9.35) reduces to the transversedynamic spin susceptibility of free electrons χ−+

0 (q, ω) [13]. At very low T , theFermi distribution reduces to the step function, so that the only contributionsto the sum over k space in (9.36) are those terms in which one state is above andthe other below the Fermi level. One can describe the energy difference ∆p,q

in the denominator of (9.36) as individual electron-hole excitation energies, theStoner excitations. Let us assume, without any loss of generality, that the metalin the ferromagnetic phase is polarized with the majority band ↓. This meansthat the average z component of spin per atom is m = n↓ − n↑ > 0. In thesimplest model of independent electrons for a metal one includes the effect ofthe periodic crystal potential approximately by substituting the free electron


mass by an effective mass (effective mass approximation) so that the electronenergy is

εk = h2k2/2m∗ (9.38)

in the simple case in which the effective mass is isotropic.In general, (m∗)

−1is a tensor [4]. Adopting the isotropic form (9.38) we

write the Stoner excitation energy (9.37) as:

∆k,q = εk+q↑ − εk↓ + ∆0 (9.39)

We keep the spin indices in the free electron energies because the split bandshave different Fermi wave vectors.

9.6.2 Singularities of the susceptibility

The poles of χ−+(q, ω) determine the excitation spectrum of electron-hole pairswith spin flipping. In the paramagnetic state, the spin change is irrelevant fromthe point of view of the energy of the excitation, at least as long as there isno external magnetic field. In the ferromagnetic state, there is a change ∆0 ofenergy with a spin flip, even though the kinetic energy change be zero in (9.39).According to (9.21), which defines the single particle Hartree–Fock energies, theFermi wave-vectors for each spin are determined by the conditions

h2k2Fσ

2m∗+ Un−σ = µ (9.40)

where µ is the Fermi level, and σ =↑ or ↓. Then, if the ground state has amajority of down spin electrons, kF↓ > kF↑.

The propagator χ−+(q, ω) has a cut along an interval of the real energy axiswhere, for fixed q, the electron-hole excitation energy ∆p,q varies between itsmaximum and minimum values, which are respectively

∆max(q) = ∆0 +h2

2m∗

(

2kF↓q + q2)

∆min(q) = ∆0 +h2

2m∗

(

−2kF↓q + q2)

(9.41)

The curves for ∆max,min(q) are represented in Fig. 9.3 for an incomplete FMand in Fig. 9.4 for a complete one.

Exercise 9.3Verify Eqs. (9.41).

There may be other singularities of χ−+ for fixed q. For some ω(q) outsidethe cut, the denominator of (9.36) might vanish, which implies that the condi-tion

Uχ−+0 (q, ω(bfq)) = 1 (9.42)


Figure 9.3: Magnon dispersion relation ωq and upper (Emax) and lower (Emin)limiting curves of the spectrum of Stoner excitations in an incomplete ferromag-netic metal within the isotropic one-band effective-mass model, shown schemat-ically as a function of q. The majority band is ↓. The hatched region is coveredas k spans all its allowed values.

is satisfied. Solutions of this equation for ω(q) are eigenvalues of the HubbardHamiltonian in the Hilbert subspace which contains an electron-hole pair ex-cited from the Fermi sea. One may interpret the corresponding eigenstate as astationary superposition of an electron and a hole of opposite spins, of energyω(q), carrying spin 1 and momentum q. This is a magnon in the present for-mulation. Eq. (9.42) requires that χ−+

0 (q, ω(q)) > 0. Let us consider the smallq limit. Then, χ−+

0 (q, ω) ≥ 0 for ω → −∞ and a look at Fig. 9.5 shows thatfor some ω > 0 Eq. (9.42) is satisfied. In that figure we depict schematicallythe variation of χ−+

0 as a function of ω for fixed q. There is a series of poles ofχ−+

0 which in the thermodynamic limit, that is N → ∞, fill the cut containedwithin the interval

ω1 = min (∆min, 0) ≤ ω ≤ ω2 = ∆max) (9.43)

spanned by all the admissible values of k, for fixed q. Ones sees that there are ingeneral as many intersections of the curve with the horizontal line χ−+

0 = 1U , as

poles , except for one, which lies at some energy ω(q) below the quasi-continuum.Clearly, when U → 0, ω(q) → ∆min(q) and the discrete state merges into thequasi-continuum. One can prove that

limq→0

ω(q) = 0 (9.44)


Figure 9.4: Same as in Fig. 9.3 for a complete FM. There is a minimum exci-tation energy δ at q = kF↓ for the creation of a single electron-hole pair.

Figure 9.5: Schematic variation of χ−+0 (q, ω), for fixed q, as a function of ω.

Eq. (9.44) is a particular case of the application of Goldstone’s theorem, andthe q = 0 magnon is a Goldstone boson. One can prove as well that the magnondispersion relation is

ω(q) = Dq2 +O(q3) (9.45)

for small q.


To be more precise:


ω(q) =h2q2

2M∗+O(q3) (9.46)

where

M∗

m∗=

[

1 − 2

3π2m2U(n↓〈ε↓〉 − n↑〈ε↑〉)

]−1

(9.47)

and we assume that the average down magnetization m = n↓ − n↑ > 0. InEq. (9.46) 〈εσ〉 is the average kinetic energy in the sub-band of spin σ.

If the majority spin, as assumed above, is down, the parenthesis in (9.47) ispositive, and M∗ > m∗. So, for a model of one parabolic band, the magnoneffective mass, as obtained within the RPA, is larger than the electron one.

It is instructive to look at this problem from another point of view, by ap-pealing to the equation of motion method. We are guided in this case by themain idea that there are excitations which consist of the correlated motion of anelectron in the spin up (minority) band and a hole in the spin down band, withtheir center of mass propagating with momentum q. We may therefore guessthat the operator for the creation of such a presumably well defined excitation,consists, to lowest order, of a sum of products of electron and hole operatorswhich create pairs with the required quantum numbers, taken from all the pos-sible free electron states, with some weight function to be determined later. Weconstruct then the operator [6]

B†(q) =∑

k

αq(k)a†k+q ↑ak ↓ (9.48)

where αq(k) is a c-number. The next step, as in the case of the AFM magnonsin Chap. 4, is to demand that this operator satisfy the Heisenberg equation

[

H,B†(q)]

= ω(q)B†(q) (9.49)

which implies that B†(q) | 0〉 would be an approximate excited eigenstate of Hwith energy E0 + ω(q). The vacuum state | 0〉 is the FM ground state of theFermi sea.

Exercise 9.5Prove that the function αq(k)is:

αq(k) =A(q)

ω(q) − (Ek+q↑ −Ek↓)(9.50)

where A(q) is a normalization constant.


The operators B(q), B†(q) have some complicated commutation relations:


[

B†(q), B(q′)]

=

[

∑

k k′

c†k′+q′↑ck+q↑δk+k′ −∑

k k′

c†k↓ck′↑δk+q,k′+q′

]

×α∗q(k)αq′ (k′) (9.51)

However, within the RPA we can substitute the number operators in (9.51) bytheir corresponding self-consistent averages, and demand that in this approxi-mation the right hand side of (9.51) is δq,q′ . This condition also determines theunknown factor A(q):


A2(q) =1

(∂χ−+/∂ω)ω(q)

(9.52)

Notice that this is precisely U 2 times the residue of χ−+ at its pole ω(q). This isjust a verification of the proportionality between the square of the wave functionand the residue of the corresponding Green’s function [14].

Consideration of the band structure details is necessary to obtain a quanti-tative description of the dispersion relation of magnons in real metallic ferro-magnets [15, 16]. The theoretical results can be compared with data obtainedwith inelastic neutron scattering[17].

9.7 Tc in Stoner model

Let us consider again a metal with only one band. The average electronic chargeand 〈Sz〉 per atom are :

n = n↓ + n↑

2〈Sz〉 ≡ m = n↓ − n↑ > 0 (9.53)

where the average occupation numbers above were defined in (9.22). When Tapproaches Tc from below m → 0. We can then expand the self consistencyconditions (9.22) in powers of m. We first invert the system (9.53), expressingn↓ and n↑ in terms of n and m:

nσ = n/2 − σm/2 (σ = ±1) (9.54)

ThenEpσ = εp + σUm/2 (9.55)

with εp ≡ εp + Un/2− µ. Let us now expand the Fermi function around εp:

f(Epσ) = f(εp) +1

2σUm

∂f

∂εp+ O(m2) (9.56)

9.7. TC IN STONER MODEL 209

We must impose1

N

∑

p

f(εp) =n

2(9.57)

Exercise 9.8Prove that the critical temperature must satisfy the condition

1 =U

N

∑

p

∂f

∂εp(9.58)

Since the integral in (9.58) is obviously bounded, and we have assumed thatthere is an ordered ferromagnetic state at T = 0, we cannot have an arbitrarilysmall U . There is therefore a minimum U compatible with this assumption, aswe already know (Stoner condition).We shall now calculate explicitly the r. h. s. of (9.58). To this end, we take thethermodynamic limit, N → ∞, and substitute consequently the sum over thequasi continuum set of wave vectors by an integration over the continuum:

U

N

∑

p

∂f

∂εp=

∫ ∞

−∞

dε ρ(ε)∂f

∂ε(9.59)

where ρ(ε) is the density of one electron states per atom in the non-magneticphase and ε = E − µ. Let us recall that

∂f

∂ε= −β ex

(ex + 1)2(9.60)

which is an even function of x ≡ βε which decreases rapidly as | x |> 1, or| ε |> kBT . For all transition metals the band width W and the Fermi level µ ,are much larger than kBT , so that the variation of ρ(ε) is very small within theinterval where the function (9.60) is non negligible. This allows us to expand ρin a Taylor series in ε around ε = 0 (Fermi level):

ρ(ε) = ρ(0) + ε ρ′(0) +1

2ε2 ρ′′(0) + · · · (9.61)

The second moment of the derivative of the Fermi distribution is [18]:

∫ ∞

−∞

ex

(ex + 1)2x2dx =

π2

3(9.62)

so that

−∫

dε ρ(ε)∂f

∂ε= ρ(0) + ρ′′(0)

π2

6β2c

+ · · · =1

U(9.63)

Orπ2

6(kBTc)

2 ≈ Uρ(0) − 1

Uρ′′(0)(9.64)


We observe that in this approximation Eq. (9.64) can only be satisfied if theFermi level is near a maximum of the density of states. We can roughly estimatethe order of magnitude of Tc as obtained from (9.64) in terms of the bandwidthW , by assuming that ρ(0) ≈ 1/W and that ρ′′(0) ≈ 1/W 3. Then we getkBTc ≈ W if U and W are of the same order, which is supposed to happen inthe 3d transition metals. This temperature is of the order of several eV , or tensof thousands K, which of course is almost two orders of magnitude larger thanthe measured Curie temperatures in the ferromagnetic metals. The conclusionis therefore that the simple Stoner model does not predict correctly the valueof Tc. The prevalent current view is that the loss of magnetization does notproceed by the overall change into the non magnetic state at each site, as inthe Stoner model, but rather that at each site there may persist a non zero neteffective magnetic moment, even for T several times Tc. The orientation in spaceof these local moments would be however random at T > Tc, thus destroyingthe long range magnetic order, and yielding a vanishing total magnetization[19, 20].

9.8 Metals with degenerate bands

Let us consider now a more realistic case in which we extend the basis of Wannierorbitals by including Nd degenerate states on each site. For instance, in a 3dtransition metal Nd = 5, corresponding to the five d = 2 orbitals. If the electronannihilation operator in second quantization is expanded in the correspondingspin-orbitals, we get

ψσ(x) =∑

α,m

Wm(x−Rα) | σ〉aαmσ (9.65)

where the wave function Wm(x−Rα) is the Wannier orbital of type m centeredat site Rα, and m = 1, · · · , Nd. The two-body interaction in this basis has theform

V =1

2

∑

σσ′

∫ ∫

dx1dx2ψ†σ(x1)ψ

†σ′(x2)V (x1 − x2)ψσ′(x2)ψσ(x1) (9.66)

If we assume as before that the only important terms in the interaction are theintra-atomic ones, we generalize Hubbard’s interaction Hamiltonian to

VHu =1

2

∑

α,σ,σ′

∑

m1m2

∑

m3m4

(m1m2 | V | m3m4)a†αm1σa

†αm2σ′aαm3σ′aαm4σ (9.67)

where

(m1m2 | V | m3m4) ≡∫ ∫

dx1dx2 W∗αm1

(x1)W∗αm2

(x2)V (x1 − x2)Wαm3(x2)Wαm4(x1) (9.68)

which is independent of α.

9.8. METALS WITH DEGENERATE BANDS 211

Let us now make some further simplifications:

1. We assume that the diagonal term is the same for all m:

(mm | V | mm) = I ;

2. A given pair of different orbitals m,m′ gives rise to a direct interactionmatrix element

(mm′ | V | m′m) ≡ U (9.69)

which we assume is the same for all possible pairs m 6= m′ and

3. an exchange term

(mm′ | V | mm′) ≡ J (9.70)

also assumed the same for all pairs m 6= m′.

4. Finally we assume that symmetry requires that

(mm | V | m′m′) ≡ 0 (9.71)

for m 6= m′.

The three parameters so defined cannot be independent. Suppose we performa symmetry operation such that the two orbitals involved are transformed intotwo linear combinations thereof, as usual. Since the Hamiltonian is invariantunder the transformation the parameters must also be invariant. For exampleconsider U → U ′, where the new orbitals are

|M〉 = c | m〉 + s | m′〉|M ′〉 = −s | m〉 + c | m′〉 (9.72)

with c2 + s2 = 1, and we take c, s =real. It is easy to show now that

(MM ′ | V |M ′M) = U ′ = U

if

I − J = U (9.73)

which we shall assume in the following. Using this relation we can write theinteraction term as

VHu =1

2(U + J)

∑

αmσ

nαmσnαm−σ

+1

2(U + J)

∑

α,m6=m′,σ

nαmσnαm′−σ

+ (U − J)nαmσnαm′σ − Ja†αmσaαm−σa†αm′−σaαm′σ (9.74)


We remark at this point that the terms m 6= m′ can be written as

(

U − J

2

)

∑

α,m6=m′,σ,σ′

nαmσnαm′,σ′ − 2J∑

α,m6=m′

Sαm · Sαm′ (9.75)

which expresses Hund’s rule that favours the maximum possible value of | Sαm+Sαm′ |. The one-electron hopping term is assumed to connect orbital stateson different sites with a matrix element which is independent of the orbitalquantum-number and of spin, so that the complete Hubbard Hamiltonian is

HHu =∑

αα′σm

tαα′a†αmσaα′mσ + h.c. + VHu (9.76)

We shall now apply the usual Hartree–Fock approximation. To this end thenumber operators are expressed identically as

nαmσ ≡ (nαmσ − 〈nαmσ〉) + 〈nαmσ〉 (9.77)

Exercise 9.9Substitute the identity (9.77) into the interaction potential, and neglect the prod-ucts of fluctuations to obtain the HF Hamiltonian

HHF =∑

αα′σm

tαα′a†αmσaα′mσ + h.c. + ( U + J )∑

αmσ

nαmσ〈nαm−σ〉

+U∑

αm6=m′σ

nαmσ〈nαm′−σ〉 + (U − J) +∑

αm6=m′σ

nαmσ〈nαm′σ〉 (9.78)

Let us now define the average spin and charge of the ion on site α:

Szα =

1

2

Nd∑

m=1

(〈nαm↑〉 − 〈nαm↓〉)

n =

Nd∑

m=1

(〈nαm↑〉 + 〈nαm↓〉) (9.79)

We are assuming that n is site independent. Then the HF interaction potentialcan be written as

VHF =n

2Nd[ (2Nd − 1)U + (2 −Nd)J ]

∑

αmσ

nαmσ

− U +NdJ

NdSz

α

∑

αm

( nαm↑ − nαm↓ ) (9.80)

As an example let us consider the d states in a transition metal, so thatNd = 5.

9.9. SPIN-DENSITY WAVE 213

Exercise 9.10Show that Stoner’s criterium for the FM instability in this case is

Ieffρ(µ) ≥ 1 (9.81)

where Ieff = U + 5J and ρ(µ) is the density of states per atom at the Fermilevel of each of the Nd degenerate (in the present approximation) bands of non-interacting electrons.

We see that the effect of degeneracy, under the present assumptions and simpli-fications, is to enhance the effective Stoner intra-atomic interaction parameterand accordingly to favour the onset of FM ordering.

9.9 Spin-density wave

Let us turn now to other possibilities than the uniform ferromagnetic instability.Just as in the Heisenberg model, we must look for static (ω = 0) instabilitiesof the response function for some finite q. To simplify the treatment of theproblem we shall return to the case of non-degenerate bands.

We first calculate the free electron susceptibility χ−+0 (q, 0). From Eq. (9.36)

we get

χ−+0 (q, 0) = − 1

N

∑

k

fk − fk+q

Ek −Ek+q(9.82)

We consider the paramagnetic phase and take the limit N → ∞, so that thesum tends to the integral

χ−+0 (q, 0) =

2V

8π3N

∫

d3kfk

Ek+q −Ek(9.83)

which in the paramagnetic phase coincides with the expression for the staticdielectric polarizability of an electron gas [21] and can be written as

χ−+0 (q, 0) =

3z

4µF (q) (9.84)

where

F (q) = 1 +

(

4k2f − q2

4qkF

)

log2kF + q

2kF − q(9.85)

and we used the relationV

Nk3

F = 3π2z (9.86)

valid for a metal of total volume V with N atomic cells, one atom per atomiccell and z electrons per atom.


The RPA dielectric function for an electron gas was first obtained by J.Bardeen [22] and independently by J. Lindhard [23] and it is known as Lindharddielectric function. Its calculation leads to Eq. (9.84) for the polarizability. Onecan easily show that F (q) defined in Eq. (9.85) is monotonically decreasing inthe interval 0 ≤ q < ∞, so its maximum is at q=0, where F (0) = 2. Then thecondition

χ−+0 (q, 0)U = 1 (9.87)

is first satisfied, as U increases from 0, at q = 0, which leads again to theuniform ferromagnetic phase. For q = 0 we Eq. (9.84) reduces to the Paulistatic susceptibility of the free electron gas

χP =3z

2µ(9.88)

as obtained in Sect. 9.2. The paramagnetic contribution in a ferromagneticmetal may have some quantitative effect upon comparison of theory and experi-ment, and it must be taken into account when one needs to fit some parametersof a model to measurements.

The description of the possible phase transitions changes considerably if arealistic band structure is considered. Take for instance a simple cubic lattice,where the kinetic part of the energy has the form

εk = −ε0(cos kxa+ cos kya+ cos kza) (9.89)

which is a band of width W = 6ε0, and with its minimum at the Γ point k = 0.The density of states of this band is symmetric around εk = 0 [24], so that if theband is half full at T = 0 the Fermi energy is µ = 0. Consider now the specialwave vector Q = π

a (1, 1, 1). We verify from Eq. (9.89) that

εk+Q = −εk ∀ k

and for the special case εk = 0, that also implies

εk+Q = εk .

This means that different points of the energy surface εk = 0, are connected byQ, which is half a reciprocal lattice vector. For this special Q we can calculateχ−+

0 (Q, 0):

χ−+0 (Q, 0) = − V

8π3N

∫ µ

−3ε0

dερ(ε)

ε(9.90)

which has a logarithmic singularity as µ→ 0. Then the instability for a pertur-bation of the magnetization with wave vector Q will occur even for U = 0 [25].This is a spin-density-wave (SDW) instability. This situation, in which differentfinite portions of the Fermi surface are connected by a particular wave vector,is called nesting.

Ground states of metals with a static spin-density-wave were predicted byA. W. Overhauser [26], who concluded that for a metal described within the

9.10. HARTREE–FOCK DESCRIPTION OF SDW 215

Hartree–Fock approximation with an unscreened Coulomb interaction (althoughnot for a real Fermi liquid at normal metallic densities), the susceptibility χ(q)diverges at q = 2kF .

The magnetization in the SDW phase that results from the instability de-scribed in the example above oscillates in space as

m(r) = m0 eiQ·r (9.91)

and for the special s. c. case it takes opposite signs on neighbouring sites in thelattice, in an AF1 spin structure, with spins aligned parallel on each plane of the(111) set, perpendicular to Q, while spins on adjacent planes are anti-parallel.As we mentioned above, in realistic cases a finite U will be necessary to producesuch an instability.

9.10 Hartree–Fock description of SDW

Let us return to the Hubbard Hamiltonian in the site representation

H =∑

i,j,σ

tija†iσajσ + U

∑

i

ni↑ni↓ (9.92)

We can express the number operator for given spin in a form which displays thelocal z component of spin:

niσ ≡ 1

2ni + σSz

i

ni ≡∑

σ

niσ

Szi ≡

∑

σ

1

2σniσ (9.93)

In the HFA we neglect as usual terms of second order in the fluctuations, leadingto

H =∑

i,j,σ

tija†iσajσ + U

∑

iσ

niσ〈ni−σ〉 − U〈ni↑〉〈ni↓〉 (9.94)

We now use Eq. (9.93) to rewrite the efective potential:

U∑

iσ

niσ〈ni−σ〉 = U∑

iσ

niσ〈1

2ni + σSz

i 〉 (9.95)

At this point we make the following two assumptions:1) The charge density is uniform, so:

〈 1

2ni 〉 =

n

2, ∀i .

2) The z component of the spin-density oscillates with a fixed wave vector:

〈Szi 〉 = SQ(eiQ·Ri + e−iQ·Ri)


We return to the running-wave representation and find the following form forHubbard’s Hamiltonian in the HFA:

H =∑

kσ

Ekσa†kσakσ

−USQ

∑

kσ

σ(

a†kσak+Qσ + a†k+Qσakσ

)

−∑

i

Un2

2+ U

∑

i

〈Szi 〉2 (9.96)

where the quasi-particle energy is

Ekσ = εkσ − σUSQ (9.97)

and we have the relations

〈Szi 〉 = 〈Sz

Q〉eiQ·Ri + c. c.

SzQ ≡ 1

2

∑

kσ

σa†kσak+Qσ

〈SzQ〉 ≡ SQ (9.98)

In summary, we have added a molecular field to the kinetic energy eachelectron had in the paramagnetic phase and we obtained a coupling of pairs ofelectrons with wave vectors k,k + Q. Before we continue it will be useful tosimplify the model somehow by assuming that

2Q = G (9.99)

where G is a reciprocal lattice basis vector. Then, we consider a domain withinthe first BZ containing, and symmetrical with respect to, the origin, that wecall RBZ (Reduced Brillouin Zone) with the property that

∀p ∈ RBZ , p + Q ∈ BZ (9.100)

in such a way that the whole set of points k inside the BZ is the direct sumof the set p and the set p + Q. In the particular case Q = (Q, 0, 0), we cantake p :| px |≤ Q/2 for the RBZ. One easily verifies that for any such p eitherp + Q or p + Q−G lies inside the BZ. Besides, due to the condition (9.99) wehave ap+2Q, σ = apσ, etc. Let us now write the sums over the BZ using thisdecomposition, where p is always inside the RBZ:

H =∑

pσ

Epσa†pσapσ

+∑

pσ

Ep+Qσa†p+Qσap+Qσ

− 2USQ

∑

pσ

σ(

a†pσap+Qσ + a†p+Qσapσ

)

+E0 (9.101)


where E0 contains the c-number part of H . Diagonalizing H so written is now afamiliar problem, and we naturally proceed to perform the necessary Bogoliubovtransformation from the pair of operators apσ , a(p+Q)σ to a new pair c1pσ , c2pσ .The transformation is parametrized conveniently as a rotation:

a†pσ = c†1pσ cos θpσ − c†2pσ sin θpσ

a†(p+Q)σ = c†1pσ sin θpσ + c†2pσ cos θpσ (9.102)

Exercise 9.11Prove that the anticommutation relations

cαpσ , c†βp′σ′ = δαβδpp′δσσ′ (9.103)

are preserved, that the non-diagonal terms of the form c1c†2 , etc. have vanishing

coefficients if:

tan 2θpσ =2USQσ

Ep+Q −Ep(9.104)

and that if θpσ is so chosen the resulting transformed Hamiltonian is:

H =∑

αpσ

Eαpσc†αpσcαpσ (9.105)

where α = (1, 2) and the new quasi-particle energies are:

E1(2)pσ =1

2[ εp + εp+Q + Un

±√

(εp − εp+Q)2 + (2USQ)2]

(9.106)

where 1(2) → −(+).

We get therefore two bands, but they are not distinguished, as in Stoner ferro-magnet, by spin, since as we see the quasiparticle energies are spin independent,and two electrons, with both spins, will occupy each of these states in the Fermivacuum, up to the Fermi level. Since the Bogoliubov transformation is uni-tary the operator for the total number of particles takes the same form in bothrepresentations:

∑

αpσ

c†αpσcαpσ =∑

kσ

a†kσakσ (9.107)

and as a consequence

∫ E2u

E1 l

dE N(E) =

∫ W/2

−W/2

dε ρ(ε) (9.108)


where E1 l, E2u are respectively the lower limit of the lower sub-band and theupper limit of the upper sub-band, N(E) is the density of states per atom perspin in the AFM phase and ρ(ε) the corresponding function for the free electron(U=0) problem, assumed to be symmetric and of width W . Remark that onlyRBZ wave-vectors are involved on the l. h. s. of Eq. (9.107), while k vectors onthe r. h. s. span the whole (atomic) BZ. One can use the periodicity of Ep,

Ep+G = Ep

to plot both AFM sub-bands as one band in the extended RBZ, which coincideswith the original BZ, or plot both sub-bands within the RBZ.

In special cases we find a gap separating both sub-bands. Let us considerthe perfect nesting case where

εp+Q = −εp , ∀ p (9.109)

This is also called the folding condition. In the case of only first n. n. hopping,one can find some vectors Q which fulfill this condition in the s. c. and b. c. c.lattices, but not in the f. c. c. one. If (9.109) holds, the sub-band energies ofthe AFM phase are

E1,2 =Un

2∓√

ε2p + (USQ)2 (9.110)

where 1(2) refers to the lower (upper) sub-band.Let us now calculate the density of states per atom, assuming ρ(ε) is known.

By definition, and using Eq. (9.110) explicitly, we have

N1,2(E) =2

N

∑

p

δ

(

E − Un

2±√

ε2p + (USQ)2)

(9.111)

Let us simplify the notation and call USQ ≡ Γ. Now we take the thermodynamiclimit and substitute the sum by an integral:

N1,2(E) = 2

∫

dε ρ(ε)δ(

(

E − Un

2±√

ε2 + Γ2

)

(9.112)

The argument of the δ distribution above has the form

X(E, ε) = E − Un

2+ f(ε)

and for fixed E the only contributions to the integral arise from the set εi ofthe roots of X . Then,

δ(X) =∑

i

δ(ε− εi)

∂f/∂ε εi

(9.113)

The roots of X are

ε± = ±√

(E − Un

2)2 − Γ2 (9.114)


We notice that there are no real roots in the interval of E

Un

2− Γ ≤ E ≤ Un

2+ Γ (9.115)

which therefore constitutes a gap of width 2Γ.

Exercise 9.12Show that the density of states per atom for both sub-bands is

N1,2(E) = 2ρ(ε±) | E − Un

2 |√

(E − Un2 )2 − Γ2

(9.116)

We can write (9.116 ) as

N1,2(E) = N (ω) = 2ρ(±ε(ω)) | ω |

ε(ω)(9.117)

in terms of

ω ≡ E − Un

2

ε(ω) ≡√

ω2 − Γ2 (9.118)

We obtained a mapping of ρ(ε) onto N (ω) which yields a symmetric distributionin ω if ρ is symmetric.

At both borders of the gap, ω = ±Γ , the denominator ε(ω) = 0, so that theAFM density of states has a square root divergence (see Fig. 9.6).

Exercise 9.13Verify that the perfect nesting condition

εp+Q = −εpholds for:

1. The s.c. lattice withQ =

π

a(±1,±1,±1)

resulting in the AF1 ordering;

2. The b.c.c. lattice withQ =

π

2a(±1, 0, 0)

leading to AF1 ordering;

3. The b.c.c. lattice with

Q =2π

a(±1,±1,±1)

resulting in AF ordering with two sublattices.


Figure 9.6: On the l.h.s. of the figure we plot the density of states functionsρ↑ and ρ↓ for the non-interacting-electron (U = 0) band vs. the kinetic energyε; on the r.h.s. we show the schematic density of states of the split band as afunction of the quasi-particle energy ω as defined in Eq. (9.118) for the AFMphase.

In order to complete the solution for the AFM split band case we need tofind the conditions for the self-consistent evaluation of the order parameter SQ.From Eq. (9.98), and upon substituting the ak, ak+Q operators by c1, c2 fromEq. (9.102) we find

SQ =1

2

∑

p,σ

σ sin θpσ cos θpσ

(

c†1pσc1pσ − c†2pσc2pσ

)

(9.119)

We now calculate the thermodynamic average of the operator in Eq. (9.119)and we substitute sin θpσ , cos θpσ by their values obtained from Eq. (9.104), toobtain the result:

〈SzQ〉 ≡ SQ =

1

2

∑

pσ

USQ〈( n1pσ − n2pσ )〉√

ε2p + (USQ)2(9.120)

where 1(2) denotes the lower(upper) sub-band and

〈nαpσ〉 = (eβ(Eαpσ−µ) + 1)−1 . (9.121)

For a metal with one electron per atom

∫ ∞

−∞

dε ρ(ε) = 1

and n = 1 is the half-filled-band case, once spin is taken into account, so forthis case the Fermi level would be exactly in the middle of the band, that is,at ε = 0 if the band is symmetric. In this case, as we have seen, the split-banddensity of states is also symmetric in ω, so for n < 1 the lower band is not


completely full of electrons of both spins. Notice that the integrand in (9.120)is spin independent, since so is the spectrum, so that the sum over spin givesjust a factor 2. In order to solve completely the problem we must determine theonly parameter still unknown, namely the Fermi level µ, from the condition thatthe total average number of electrons per atom must coincide with the inputvalue n. Using Eq. (9.116) we write this condition for the case in hand as

2

∫

dωρ(√ω2 − Γ2) | ω |√ω2 − Γ2

= n (9.122)

The system of Eqs. (9.120) and (9.122) determines the two parameters SQ andµ for given n < 1. We remark that

| ω |√ω2 − Γ2

=dε

dω(9.123)

which is just the jacobian for the change of variables from ω to ε, so one cansimply rewrite Eq. (9.122) as

2

∫ εF

−W/2

dε ρ(ε) = n (9.124)

where

εF = −√

(µ− Un

2)2 − Γ2 (9.125)

and simplify also Eq. (9.120) as

1

U=

∫ εF

−W/2

dερ(ε)√ε2 + Γ2

. (9.126)

If n > 1 the lower sub-band is full, and the last two equations are substitutedby

2

∫ εF

0

dε ρ(ε) = n− 1 (9.127)

and1

U=

∫ 0

−W/2


−∫ εF

0


. (9.128)

with

εF =

√

(µ− U n

2)2 − Γ2 (9.129)

We remark that the first integral in (9.128) is independent on n.More refined treatments of the effect of nesting which include considering

several bands and a more realistic description of the Fermi surface can be foundin the literature [29].

Among the pure 3d metals ony Cr and α − Mn are AF. Body centeredCr has a sinusoidal spin-wave with Q ≈ 0.96 × 2π/a below Tc = 310 K. The


polarization is longitudinal, i.e. ‖ Q for T < 115 K and transverse (⊥ Q) forT > 115 K. The maximum value of the magnetization along the wave at low Tis 0.59 µB . Theoretical analysis based on the itinerant model, including detailedinformation on the Fermi surface has been reasonably succesful in describing theSDW in Cr [28, 27]. In real cases, the overlap of different bands will not allow ingeneral the presence of an AF gap over the whole BZ. In a realistic calculationfor Cr in the SDW phase the DOS does not vanish anywhere near the Fermisurface, although it shows a pronounced dip therein.

For further detailed treatments of itinerant magnetism the reader is referredto several reviews in the literature [28, 30, 31].

9.11 Effects of correlations

Let us return to the Hubbard Hamiltonian in the Wannier representation. Wewant to consider the limit 0 <| t | /U 1, where we shall see that in the half-filled band case n = 1 the ground state is an AF insulator.[33, 34, 35] In thislimit the hopping term in H can be treated as a perturbation. The HF solutionyields in the paramagnetic case just one band centered at E0 +Un/2. Hubbard[5] used a Green’s function decoupling approximation. In this approach onedefines the usual retarded Green’s function

Gσij(ω) = 〈〈 ciσ ; c†jσ 〉〉ω (9.130)

which obeys the equation

ωGσij(ω) = δij +

∑

l

tilGσlj(ω) + U Γσ

ij(ω) (9.131)

introducing the new Green’s function

Γσij(ω) = 〈〈 niσciσ ; c†jσ 〉〉ω (9.132)

The equation for Γ is now approximated by factoring out 〈ni−σ〉. Fourier trans-forming to k space one gets the pair of equations

(ω − εk)Gσk (ω) − U Γσ

k (ω) = 1

−〈n−σ〉εkGσk (ω) + (ω − U)Γσ

k (ω) = 〈n−σ〉 (9.133)

Now we find two real poles for each σ:

ωσ± =

εk + U

2±

√

(εk − U

2

2

+ n−σUεk (9.134)

In the PM phase one gets two sub-bands. In the narrow band limit U εk theexpression for G simplifies:

Gσk (ω) ≈ 1 − n−σ

ω − εk(1 − n−σ)+

n−σ

ω − U − n−σεk(9.135)

9.11. EFFECTS OF CORRELATIONS 223

For a spin σ electron , the original non-interacting band is split into two sub-bands. The lower one can be interpreted as originating in the electrons withspin σ which itinerate occupying succsessively ions which are empty. The widthof this sub-band is renormalized by the factor (1 − n−σ), reflecting the averagereduction of the hopping probability due to the exclusion of a fraction n−σ ofavailable sites.

The upper sub-band, on the other hand, contains itinerant states in whichthe electron occupies precisely those ions already singly occupied (by a −σ spinelectron), so the factor n−σ renormalizes the effective width of this sub-band.Since all sites contributing to this band are doubly occupied, their energy isshifted up by an amount equal to the intra-atomic Coulomb repulsion U .

If U → ∞ one expects that the ground state of Hubbard’s Hamiltonian beFM.

In this respect we have the Nagaoka–Thouless theorem [32].Nagaoka makes the following hypotheses:

1. The hopping term has only first n. n. range;

2. The band is almost half filled, so that if Ne is the total number of electronsand N that of atoms in the sample, he assumes that | Ne −N | N .

Let us consider separately two cases depending on the crystal structure:

A) s. c. and b. c. c.; f. c. c. and h. c. p. with Ne > N ;

B) f. c. c. and h. c. p. with Ne < N .

Then, for the special case Ne = N − 1, that is, exactly one hole in the band, inthe limit U → ∞ he finds that:

1. for case A the saturated (also called complete) FM state is the groundstate;

2. for case B the one above is not the ground state, which is instead anun-saturated (incomplete) FM.

We see that the U → ∞ limit is quite different from the large but finite Ucase. In the latter case, the non-interacting (U = 0) band splits into two wellseparated bands. If n = 1, at very low T the lower band is completely full andthe upper one completely empty, with a gap in between, so that we have aninsulator. It can be proven that there is an effective AF coupling which resultsin an insulating AF , which is known as a Mott insulator, being the groundstate in this limit. The AF effective exchange interaction has the structureof the Heisenberg spin Hamiltonian with an effective exchange constant of theorder of t2/U . This is known as the “kinetic exchange” interaction, which weshall now discuss.


9.11.1 Kinetic exchange interaction

We consider again the narrow band limit t/U 1, in which the hopping termin H can be considered as a perturbation.

We shall perform a canonical transformation on the Hilbert space that willeliminate the transformed hopping terms to first order and such that the groundstate and the low excited ones belong to the subspace with single or zero siteoccupation. This is to be expected because every doubly occupied site adds alarge positive contribution U to the energy [36]. First of all we shall describethe Hamiltonian as a sum of bond (instead of site) terms, so we write

H =∑

〈ij〉

H ij (9.136)

where

H ij = tij∑

σ

a†iσajσ + h.c. +U

z(ni↑ni↓ + nj↑nj↓) (9.137)

We have divided the interaction term by the coordination number z to compen-sate the overcounting, since each site contributes to z bonds. The advantage ofthis site ↔ bond transformation is that we can start by looking at the simplerproblem of a two-site, or one-bond, system first and then extend the result tothe whole system.

Let us take two ions i, j and assume tij = real, so that the bond Hamiltonianis

H(12) = t∑

σ

( a†1σa2σ + a†2σa1σ ) +U

z

2∑

i=1

ni↑ni↓ (9.138)

Let us now perform the canonical transformation

H(12) = e−iεSH(12)(ε)eiεS (9.139)

where we define

H(12)(ε) = H0 + εH1 (9.140)

with a decomposition of H(12) to be defined below. As usual in the perturbationformalism, the parameter ε will be substituted by 1 at the end of the calculation,and it only serves the purpose of keeping track of the order of each contributionto the energy. As a preliminary of what follows we shall define several projectionoperators, that will enable us to separate single from double occupation states.

Let us consider the polynomial in the variable x

∏

σ

( (1 − niσ) + xniσ ) =

2∑

mi=1

p(i,mi)xmi (9.141)

It is easy to see that the operators p(i,mi) project a state onto the subspace inwhich

∑

σ niσ = mi.


These operators have the following algebraic properties:

2∑

mi=1

p(i,mi) = 1 (9.142)

p(i,mi)p(i, ni) = p(i, ni)δmi,ni(9.143)

Exercise 9.14Obtain the explicit form of p(i,mi) and prove properties (9.142) and (9.143).

For the two-atom problem we define now the projection operators over the zeroplus single occupation subspace, P1, and on the double occupation subspace,P2, as:

P1 = p(1, 0)p(2, 0) + p(1, 1)p(2, 1) +∑

i6=j

p(i, 1)p(j, 0) (9.144)

andP2 = p(1, 2)p(2, 2) +

∑

i6=j

p(i, 2) ( p(j, 0) + p(j, 1) ) (9.145)

with the algebra

P1 + P2 = 1

PiPj = Pi δi,j (9.146)

We call H(12) = H in the following, since there is no possibility of confusionwith the complete H . We study now the effect of these projection operators onH :

Exercise 9.15Show that when these projection operators are applied on H from both sides weget:

P1HP1 = t∑

σ;i6=j

(1 − ni−σ)a†iσajσ(1 − nj−σ)

P1HP2 = t∑

σ;i6=j

(1 − ni−σ)a†iσajσnj−σ

P2HP1 = t∑

σ;i6=j

ni−σa†iσajσ(1 − nj−σ)

P2HP2 = t∑

σ;i6=j

ni−σa†iσajσ nj−σ +

U

z

∑

i

ni↑ni↓ (9.147)


We interpret P1HP1 as generating the lower sub–band, since it does not containany intra-atomic contribution to the energy. An electron with spin σ travelsonly on ions which contain no −σ electron. Conversely, P2HP2 represents theupper band, since in this case that electron will visit ions which already containanother one with opposite spin, which contributes U to the energy. Clearly,PiHPj 6=i are cross-sub-band terms.

In the atomic limit (small ε) we expect that the upper and lower sub-bandsdo not mix, so we define the terms in the decomposition of Eq. (9.140) as

H0 = P2HP2 + P1HP1

H1 =∑

i

PiHPj 6=i (9.148)

Upon expanding the exponential operator e±iεS in Eq. (9.139) in series of powersof ε we find:

H(ε) = H0 + ε ( H1 + i [ H0, S ] ) +

1

2ε2 ( 2i [ H1, S ] − [ [ H0, S ] , S ] ) + · · · (9.149)


The linear term in (9.149) vanishes if

H1 + i [ H0, S ] = 0 (9.150)

If we substitute this condition in Eq. 9.149 and let ε = 1 we get:

H ≡ H(1) = H0 +i

2[ H1, S ] (9.151)

We now substitute the definitions of H0 and H1 from (9.148) into Eqs. (9.150)and (9.151), and obtain the following equations:

P1HP2 + P2HP1 + i [ (P1HP1 + P2HP2 ) , S ] = 0 (9.152)

and

H = P1HP1 + P2HP2 +i

2[ (P1HP2 + P2HP1 ) , S ] (9.153)

Applying the projectors on both sides of (9.152) we find

PµHPν(1 − δµν) + iPµHPµPµSPν − iPµSPνPνHPν = 0 (9.154)

AssumePνSPν = γPν (9.155)


where γ is a c-number. This satisfies Eq. (9.154) for the diagonal terms. Forµ 6= ν we have

PµHPν + iPµHPµPµSPν − iPµSPνPνHPν = 0 (9.156)

Let us define the quantities Eu , El as the averages of respectively the upperand lower sub-band energies. In the narrow band (large U/t) limit we have

Eu −El ≈ U/z .

Upon substituting the diagonal terms PνHPν , (ν = 1, 2) in Eq. (9.156) bytheir averages one finds approximately

P1SP2 = −i zUP1HP2 (9.157)

P2SP1 = iz

UP2HP1 (9.158)

If we now substitute the diagonal and non-diagonal terms involving S from(9.155) and (9.158) into (9.151) we get for the transformed Hamiltonian of thetwo-site system:

H = P1HP2 + P2HP1 −z

U( P1HP2HP1 − P2HP1HP2 ) (9.159)

which, upon writing explicitly the expressions for the projectors Pi yields

H = t∑

i6=j,σ

(1 − ni−σ)a†iσajσ nj−σ)

+t∑

i6=j,σ

ni−σa†iσajσ(1 − nj−σ + i

U

z

∑

i

ni↑ni↓

−z t2

U

∑

i6=j,σ

[ niσ(1 − ni−σ)nj−σ(1 − njσ)

− niσni−σ(1 − njσ)(1 − nj−σ) ]

+z t2

U

∑

i6=j

(

S+i S

−j + S+

j S−i − 2a†i↑a

†i↓aj↑aj↓

)

(9.160)

If we sum the above two-site (or bond) Hamiltonian over all bonds we get agood approximation to the total H. If one returns to the sum over sites insteadof bonds the form (9.160) can be extended to all sites. As we sum Eq. (9.160)over all bonds, there appear some terms which involve three-center hoppings,which are longer range anyway, and should be neglected consistently with theapproximation of first n.n. hopping [36]. In the large U limit at low T the uppersub-band is empty, so that we can neglect the terms niσni−σ . The final resultfor the canonically transformed Hamiltonian in this case can be convenientlyexpressed introducing new fermion operators

biσ = (1 − ni−σ)aiσ (9.161)


in terms of which the transformed Hamiltonian reads:

H = t∑

i6=j,σ

b†iσbiσ +z t2

U

∑

i6=j

(S+i S

−j + S−

i S+j ) − z t2

2U

∑

i6=j,σ

νiσ νj−σ (9.162)

where νiσ = b†iσbiσ . The operators biσ and b†iσ satisfy the anti-commutationrelations

biσ , b†jτ = (1 − ni−σ)δijδστ

biσ , bjτ = b†iσ , b†jτ = δijδστ (9.163)

It is easy to verify that

∑

i6=j,σ

νiσ νj−σ =∑

i6=j,σ

νiσ νjσ −∑

i6=j

Szi S

zj (9.164)

In the case of an exactly half-filled band, where

ni↑ + ni↓ = 1 (9.165)

we have the Hamiltonian for Mott’s insulator,

Heff = t∑

i6=j

b†iσbjσ +z t2

U

∑

i6=j

Si · Sj (9.166)

also known as the ”t-J” Hamiltonian. Comparing Eq. (9.166) with the Heisen-berg hamiltonian, we see that the effective exchange interaction constant isnegative, leading to AF behaviour at low T .

The problem of strongly correlated Fermion systems has particular rele-vance since the discovery of the high-Tc superconducting compounds. See Refs.[37, 38] for reviews and further references.

9.12 Paramagnetic instability and paramagnons

There are many metallic systems which are close to satisfying Stoner’s condition(9.27), that is, for which

α ≡ Uρ(EF ) ≈ 1 (9.167)

with α < 1, so that the local Coulomb repulsion is not large enough to inducethe transition to the FM ground state. It can be shown that as α increasesand approaches 1 some properties exhibit singular behaviour. Doniach [13, 39]calculated the enhancement of the inelastic neutron scattering cross-section asa result of this paramagnetic instability. It is also found that the effective massof the conduction electrons, as obtained from the low-temperature specific heat,diverges at the critical point α = 1, while near the critical point the effectivemass renormalization is large and the temperature dependence of the specificheat has a logarithmic singular term.[40] This effects are important in near

9.12. PARAMAGNETIC INSTABILITY AND PARAMAGNONS 229

ferromagnetic metals such as Pd and also in liquid He3[13, 40, 46].We shall now calculate the low temperature specific heat correction due to

Hubbard U interaction. This requires the evaluation of the correction to thefree energy, which we shall perform now.

First we remind ourselves of the so called Feynmann–Helmann theorem1.Consider a Hamiltonian containing a non-interacting part H0 and an interactionpotential which is multiplied by a parameter λ as customary in perturbationtheory:

H(λ) = H0 + λV (9.168)

with eigenkets and eigenvalues

H(λ) n λ〉 = E(n λ) n λ〉 (9.169)

Call E(0 λ) ≡ E(λ) = ground state energy. We calculate now the derivative

∂E(λ)

∂λ≡ ∂〈0 λH(λ)0 λ〉

∂λ=

E(λ)∂〈0λ 0λ 〉

∂λ+ 〈 0λ | V | 0λ 〉 (9.170)

and the first term in the second line of (9.170) is ≡ 0 since by assumption theeigenkets are normalized to unity for all λ. Therefore,

∆E =

∫ 1

0

〈V 〉λ dλ ≡∫ 1

0

〈λV 〉λλ

dλ (9.171)

is the change of the ground state energy after the full interaction is switched on.A similar formula can be obtained for the change of the free energy due to

the interaction. To see this consider the definition of Helmholtz free energy ( atconstant volume):

F (λ) = −kBT lnTrρ(λ) (9.172)

whereρ(λ) = e−β(H0+λV )

Now we differentiate (9.172) with respect to λ and we find, by the same tokenas for the ground state energy, that:

∂F (λ)

∂λ= 〈V 〉λ (9.173)

where 〈·〉λ is the statistical average with the density matrix ρ(λ). Remarkthat there is no ambiguity with respect to the order of the operators in (9.173)because we are taking the trace.

Upon integrating (9.173) we find :

F (1) − F (0) ≡ ∆F =

∫ 1

0

d λ

λ〈λV 〉λ (9.174)

1According to V. Weisskopf, as cited by D. Pines in “The many-body problem”, this formula

is due to W. Pauli.


We are interested in the case where V is Hubbard local repulsion as in Sect.9.2. We can now relate the average in (9.174) to the particle-hole propagatorwe defined in (9.35) :

Exercise 9.17Prove, with the help of the fluctuation-dissipation theorem, that

〈λV 〉λ =−2

N

∑

q

∫

dω

2π

λUIm(χ−+(q, ω))

eβω − 1(9.175)

Upon substitution in Eq. (9.175) of the expression for χ−+(q, ω) which wasgiven within the RPA in (9.36), we are led to calculate the function

Φ(q, ω) = Im

∫ 1

0

d λ

(

χ−+0 (q, ω)U

1 − Uλχ−+(q, ω)

)

=

Im ln (1 − Uχ−+0 (q, ω) (9.176)

where

χ−+0 (q, ω) =

1

N

∑

p

fk+q↑ − fp↓ω + iη −Ek+q↑ +Ek↓

(9.177)

Notice that we added an infinitesimal positive imaginary part to ω in (9.177) toreproduce the correct analytical properties of a retarded propagator.We need now to calculate χ−+

0 (q, ω), which is proportional to the transversesusceptibility of the non-interacting Fermi gas (Sect. 9.5.1). We are interestedin the low T limit, and this implies that it suffices to consider the small q andsmall ω limit. Therefore we can expand fk+q in a series in powers of

∆k,q ≡ Ek+q −Ek .

Let us call

ζ = k · q/ k q

which is the cosine of the angle k makes with the direction of q. Then,

∆k,q = Eq + 2ζ√

Ek Eq

We shall ignore the spin dependence of the electron energies since we are as-suming that the system is in the paramagnetic phase, so that the exchange gapis zero. We have:

χ−+0 (q, ω) = − 1

N

∑

k

1

∆ − ω

(

f ′∆ +1

2f ′′∆2 +

1

6f ′′′∆3 + · · ·

)

(9.178)

We must consider up to the third order term above in order to get correctly thecontribution to the free energy to second order in T at low T . We can rewrite


the first three terms of Eq. (9.178) as:

χ−+0 (q, ω) = −

∫ ∞

0

dE ρ(E)1

2

∫ 1

−1

d ζ

[

f ′(E)

(

1 +ω

∆ − ω

)

+1

2f ′′

(

∆ − ω +ω2

∆ − ω

)

+1

6f ′′′

(

∆2 + ∆ω + ω2 +ω3

∆ − ω

)

+ · · ·]

(9.179)

The factor 12 in front of the ζ integral above is due to the fact that in the

definition of the band density of states ρ that integral was already performed.In order to proceed we perform first the ζ integral, to find:

χ−+0 (q, ω) = −

∫ ∞

0

dE ρ(E)

[

f ′(E)

×(

1 +ω

4√

E Eq

ln

(

2√

E Eq +Eq − ω − iη

−2√

E Eq +Eq − ω − iη

)

+1

2f ′′(E)

(

Eq + ω + O(ω2))

+2

9f ′′′(E)

(

E Eq + O(E2q , ωEq , ω

2))

+ · · ·]

(9.180)

At low T we approximate

f ′(E) ≈ −δ(E −EF ) (9.181)

which neglects corrections of the order of (kBT/EF )2, and we use the definitionof the derivatives of Dirac’s δ distribution:

∫

d x δ(n)(x)g(x) = (−1)n g(n)(0) (9.182)

With some algebraic operations [41] we arrive at:

χ−+0 (q, ω) = χP

[

1 +ω

4√

E Eq

(

iπ + O(Eq − ω√

E Eq

)

− 1

2

ρ′(EF )

ρ(EF )

(

Eq + ω + O(ω2))

+4

9

ρ′(EF )

ρ(EF )

(

Eq + O(E2q , ωEq , ω

2) + · · ·)

]

(9.183)

The absolute value of χ−+0 (q, ω) is maximum for

q2 ≈ ω√2α


when α and q/kF are both very small. Therefore we can limit the calculationsto the range ω qkF 1. Then Eq.(9.183) can be written as:

χ−+0 (q, ω ≈ χP

(

1 − Eq

36EF+ i

πmω

2qkF

)

(9.184)

We can now substitute this expression in Eq. (9.176)for Φ(q, ω) and finally insertthe result in the equation for the change in free energy , Eq.(9.174):

∆F =2

N

∑

q

∫

dω

2πn(ω) arctanφ(q, ω) (9.185)

where

φ(q, ω) =U Imχ−+

0 (q, ω)

1 − UReχ−+0 (q, ω)

=K ω

q vF

α+ Uρ(EF )(

q6kF

)2 (9.186)

and the coefficient K is

K =Uπρ(EF )

2

9.12.1 Paramagnon contribution to the specific heat

We differentiate Eq. (9.185) with respect to T , to obtain: [13, 41]

∆C(T ) =∂∆F (T )

∂T=

2

NkB

∑

q

∫

dω

2πarctanφ(q, ω) β2 ω n(ω) (1 + n(ω))

(9.187)For low T , which implies small q and ω, we can expand arctanφ in a Taylor

series in powers of φ , and keep up to the second non vanishing term:

∆C(T ) ≈ 2

NkB

∑

q

∫

dω

2π

(

φ(q, ω) − 1

3φ(q, ω)3

)

β2 ω n(ω) (1 + n(ω)) (9.188)

Let us perform first the ω integral arising from the linear term in φ: , which we

call I(1)ω :

I(1)ω =

(

k2B K

2πqvF

)

1

α+ Uρ(EF )(q/6kF )2

∫ ∞

−∞

d x x2n(x) (1 + n(x) ) (9.189)

The integral above can be obtained from the formula [42]:

∫ ∞

0

d x x2 ex

(ex − 1)2= 2Γ(2)ζ(2) (9.190)

Therefore, the first order correction to C(T ) is

∆C(1)(T ) =2

N

∑

q

AT

q

1

α+ Uρ(EF )( q6kF

)2(9.191)


We impose an upper cutoff on the q integral, since q is restricted to the rangeq ≤ 6kF . Since α→ 0 the dominant term in (9.191) is

∆C(1)(T ) ∝ T ln α (9.192)

to be compared with the specific heat of non-interacting electrons at low T ,[18]

C0(T ) = γ0m∗

mT (9.193)

where m∗ ,m being respectively the effective band mass and the free mass ofthe electron. The coefficient in (9.193) is:

γ0 =k2

B

3(6π2)1/3n−2/3m (9.194)

so that the effect of the interaction near the quantum critical point α = 0 is alogarithmic singularity of the renormalization coefficient of the electron effectivemass.

We consider now the contribution of the cubic term in (9.187):

∆C(3)(T ) = B∑

q

∫

dω

2πhω4 1

q3[

α+ Uρ(EF )/2(q/vF )2]

(9.195)

We remind ourselves of the condition ω/qvF 1, which sets a lower cutoff forthe q integral:

1

N

∑

q

1

q31

α+ Uρ(EF )/2(q/vF )2≈ − 1

2π2 nln

hω

kF vF(9.196)

Inserting this into (9.195) the dominant term at low T has the form [41]:

∆C(3)(T ) ≈ Const · T 3 ln kBT/hkF vF (9.197)

Collecting both terms we have

∆C(T ) = C0(T ) (F + g(T )) (9.198)

here F ∝ ln α is the renormalizatrion factor we found above (see Eq. (9.192)),while at low T :

g(T ) = A(T/TF )2 lnT/TF +B(T/TF )2 + C(T/TF )4 + · · · (9.199)

The quadratic and further terms arise from the 5th and successive powers inthe series in (9.187).

Equation (9.199) can be compared with the corresponding correction to theelectronic specific heat arising from their interaction with phonons. It was found[43, 44, 45] that, up to second order in the interaction, the correction has alsothe form of Eq. (9.198), where F is a mass renormalization coefficient and the


function g(t) has exactly the same form, with different coefficients and with TF

substituted by Θ = Debye temperature.As in the phonon case we can interpret this effect as the result of the virtual

emission and reabsorption by an itinerant electron of a paramagnon (in the placeof a phonon), which results in a correction to the quasi-particle energy, andconsequently in a renormalization of the effective electron mass. Alhough theparamagnon is not a well defined excitation of the system, χ0 can be interpretedas the paramagnon propagator, as we have seen, in the small ω, small q region,with ω/(qvF ) 1 where it has the maximum amplitude. Near this peak thedecay time of the paramagnon is long [13].

In order to study how real systems approach the QCP it is necessary to studythe variation of some key properties, like the low-T specific heat, as α changes.This can be done either by changing the composition of a metallic alloy or byapplying pressure to a system with a fixed composition [47].

9.13 Beyond Stoner theory and RPA

Spin-fluctuation theories have appreciably improved the understanding of itiner-ant magnetism in transition metals. However, the RPA approximation describedabove does not lead to a Curie-Weiss law. Murata and Doniach [48] and alsoMoriya and Kawabata [49] obtained this law by extending the RPA to includemode-mode coupling among spin-fluctuations (See the review by Moriya) [31].

Experiments point to the permanence of local moments above TC in tran-sition metal ferro- and antiferromagnets. This has led to formulations with abasis of both 3d and 4s states, so that both the local and the itinerant char-acter of the electron states are present from the start. Any theory of itinerantmagnetism must tackle the simultaneous consideration of correlated charge andspin fluctuations. Since the 70’s the approach to this problem was based onfunctional integral algorithms for calculating the partition function [50, 51].

Ongoing research is oriented to the application to itinerant magnetism ofextensions of the density functional theory. We refer the reader to some re-cent papers on the magnetism of transition metals and to the references therein[52, 59, 61, 62].

9.14 Magnetism and superconductivity

Some early theoretical work led to the conclusion that strong FM spin-fluctuationswould suppress the transition to a superconducting state.[46] However it hasbeen found recently that both phases can coexist, as for instance in UGe2[54, 55],ZrZn2[56] or URhGe[57].

The paramagnon mediated model of superconductivity [58, 59] is the maincurrent theory of the coexistence of both phenomena and it has predicted itsocurrence in ZrZn2 [60].

9.14. MAGNETISM AND SUPERCONDUCTIVITY 235

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Chapter 10

Indirect Exchange

10.1 Introduction

Let us now turn our attention to other systems, metals or semiconductors, inwhich spins can be considered well localized on some or all the sites of the lat-tice, while a part of the electrons are in delocalized states belonging to one ormore conduction bands.

As the first example, we have the rare earths, with incomplete 4f shells, andthe actinides, with incomplete 5f shells.

The diluted magnetic alloys constitute a second family of such systems, inwhich a non-magnetic host metal contains a small atomic percentage of substitu-tional impurities, consisting of atoms with some incomplete shell, with unpairedspins, which maintain, even when dissolved in the host metal, a spontaneousmagnetic moment.

The third family to consider are the magnetic semiconductors.In the first two cases, the f shell wave functions are tightly bound, and con-

sequently well localized in space. Neutron studies reveal that the mean radius,for trivalent Nd3+ and Er3+ is ∼ 0.35 A[1].

In the whole series of the trivalent lanthanide (rare earth) group, only Gdhas L = 0, an S ground state term. For all other ions in the series L 6= 0 andwe can adopt, as discussed in Chap. 1, the Russel-Saunders coupling of L toS. As already mentioned in Chap. 1, crystal field effects are much smaller thanspin-orbit effects in these elements, so that as a first approximation we considera ground level degenerate multiplet determined by Hund’s rules, with degener-acy 2J + 1 for given J . The higher multiplets in the rare earths are sufficientlyseparated in energy to be neglected. A typical value of the spin-orbit splittingparameter Ah2 is 0.01 eV [2]. Neglecting crystal field effects then, we have

µ = gJµBJ (10.1)

For the second half of the lanthanide series, that is for Tb, Dy, Ho, Er, Tm andY b, parallel coupling of L and S is required by Hund’s rules, that is J = L+S,

239

240 CHAPTER 10. INDIRECT EXCHANGE

and then

gJ =L+ 2S

L+ S(10.2)

which impliesS = (gJ − 1)J (10.3)

This has been experimentally verified for all ions listed above [3]. Static suscepti-bility measurements in these metals also confirm the Curie free-ion paramagneticbehaviour at temperatures well above the crystal field splitting:

χ0 = (gJµB)2J(J + 1)

3kBT(10.4)

supporting the view that the 4f spins are localized, and behaving as free ionsin the paramagnetic phase (T Tc) (Compare with Eq. (1.6)).

As a consequence of the negligible overlap among 4f orbitals on differentsites, magnetic cooperative effects in these metals depend upon the coupling ofspins through the agency of the conduction electrons.

We have already seen that within the Heitler-London approximation, themain factor leading to spin dependent interactions among electrons on differentatoms is the exchange integral j (Eq. (2.28)), involving the Coulomb electro-static interaction potential. In the problem considered in Chap. 2, we were con-cerned with two atoms, a and b, which could exchange electrons. In the presentcase we shall consider exchange processes involving two types of states:localizedstates, either d or f , on the one hand; on the other hand, itinerant delocalizedBloch states, originating in predominantly s and p atomic states with apprecia-ble overlap belonging to one or more conduction bands, which are reasonablywide (several eV).

10.2 Effective s-d exchange interaction

We construct Slater determinants as a convenient basis for the Hilbert spaceof the many electron system from anti-symmetrized products of wide band sand p states and narrow band d or f localized states. In the simplest case weconsider only two kinds of states. One is fairly localized within a Wigner-Seitzcell around a given site, which we shall call a “d” state and which we takeas a singlet for simplicity. A second set of states, which we shall call “s” dooverlap considerably, and form a conduction band. We expand the matter-waveoperators Ψσ(x),Ψ†

σ(x) in the basis of the d and s orbitals (we may assume thatthese orbitals are mutually orthogonal), so that we write

Ψσ(x) '∑

nσ

φn(x)dnσ +∑

kσ

akσψkσ(x) (10.5)

The Coulomb interaction in this representation is

VC =1

2

∫ ∫

dr dr′∑

σσ′

Ψ†σ(r)Ψ†

σ′ (r′)v(r − r′)Ψσ′(r′)Ψσ(r) (10.6)

10.2. EFFECTIVE S-D EXCHANGE INTERACTION 241

where v(r − r′) = e2/ | r − r′ | . Upon substitution of (10.5) into (10.6) therewill be 8 types of terms, which can be schematically represented as

1. d†d†dd

2. c†c†cc

3. d†d†cc

4. c†c†dd

5. d†c†cd

6. d†c†dc

7. c†d†cd

8. c†d†dc

The terms involving exchange are (3), (4), (6) and (7). Among these (3) and(4) involve two electrons on a d orbital. If the orbitals are on the same site, thisintegral is of the order of the intra-atomic U for the d orbitals, by assumptiona large quantity, so that the corresponding energies are outside the range of theexcitations we want to describe. In the perturbation expansion of the energythat we shall perform below, these states will contribute terms with very largedenominators, so that we shall neglect them altogether. If both d orbitals areon different sites, they are orthogonal by hypothesis, and we also neglect theseterms.

We are then left with exchange terms (6) and (7) which involve the exchangeof an electron between a d and an s state. It is easy to see that both terms areequal, since we can interchange them by renaming the integration variables inthe expression for the matrix element. We shall consider only the on-site termsn = m among all products involving two d, d† operators , since by assumptionthe d orbitals are well localized.

The conduction band s wave functions are Bloch states

ψk(r) =1√Nuk(r)eik·r (10.7)

where uk(r)eik·r has the periodicity of the lattice and is normalized within acrystal cell, so that ψk(r) is normalized over the whole crystal. If we add togetherterms (6) and (7) we get

Hsd = − 1

N

∑

nσσ′

∑

kk′

Jkk′ei(k−k′)·Rn d†nσdnσ′a†kσ′ak′σ (10.8)

where the exchange integral Jkk′ is

Jkk′ =

∫

dr

∫

dr′φ∗(r)u∗k(r′)e−ik·r′φ(r′)uk′(r)eik′ ·rv(r − r′) (10.9)


Exercise 10.1Show that leaving aside a term which is spin-independent the other terms in(10.11) can be cast in the form [5]

Hsd = − 1

N

∑

n

∑

kk′

Jkk′ei(k−k′)·Rn Szn( a†k↑ak′↑

− a†k↓ak′↓ ) + S+n a

†k↓ak′↑ + S−

n a†k↑ak′↓ (10.10)

If we substitute the Bloch states by plane waves, and write explicitly theCoulomb potential, the expression for the exchange integral above simplifies(we call k′ = k + q ):

J(k,q) = e2∫

dr1dr2eik·r1e−i(k+q)·r2φ(r1)φ(r2))

| r1 − r2 | (10.11)

More complicated (and realistic) cases in which the orbital degeneracy of thelocalized orbitals is considered, as well as the possibility of core states withseveral electrons, are also considered in the literature [6]. We shall restrictourselves to the simple form above for the s−d interaction We consider now theconduction electrons as independent particles and proceed, following Yosida [5],to calculate in perturbation theory the effect of Hsd on the total energy of theconduction band electrons, considered as a Fermi gas. The first order correctionto the energy of the s electrons is given by the expectation value of Hsd in theFermi sea ground state:

∆E(1) = − 1

NJ(0) (〈n+〉 − 〈n−〉)

∑

n

< Szn > (10.12)

where z is the spin quantization axis and 〈n±〉 are the total numbers of conduc-tion electrons of spin + and −. We assume that J(k, q) above can be approxi-mated by a constant J0. Assuming J0 > 0, (10.12) implies, if 〈Sz

n〉 > 0, that tominimize the energy to this order one would obtain a net uniform polarizationof the conduction electrons, with 〈n+〉−〈n−〉 > 0. We denote with n± the totalnumber of electrons with the corresponding z component of spin.If the resultant net polarization in (10.12) is ↑, then for an isotropic bandkF↑ > kF↓. The expectation value of the total energy is:

Ekin + ∆E(1) =∑

σ

kF σ∑

k

[

εkσ − σJ(0)

N

∑

n

< Szn >

]

nkσ (10.13)

with an obvious notation. In this chapter the equivalent notations ± or ↑, ↓ willbe used for spin, according to convenience.



The Fermi level µ in the presence of the perturbation must be the same forboth spins, so that

µ = εkF↑− σJ(0)

N

∑

n

< Szn >= εkF↓

+σJ(0)

N

∑

n

< Szn > (10.14)

The total number of conduction electrons of a given spin satisfies the relation

nσ =

(

V

6π2

)

k3Fσ (10.15)

Consider the perturbation of kF of the paramagnetic state when the perturba-tion is present. From (10.15) we see that, to first order, we can write

kFσ = kF ± ∆kF (10.16)

with

kF =1

2(kF↑ + kF↓)

∆kF =1

2(kF↑ − kF↓) (10.17)

To the same order, we have

µ = EF ± 2EF∆kF

kF(10.18)

where EF is the unperturbed Fermi level.The total number of conduction electrons, which we call 2n for convenience, is

2n = n+ + n− (10.19)

so that

n =V

6π2k3

F +O(

(∆kF )2)

(10.20)


Eq.(10.20) the conservation of the number of electrons (or charge) to first order.

Exercise 10.4Prove the following relations to first order:

2∆kFEF

kF= J(0)M (10.21)


and

nσ = n+ 3σJ(0)Mn

2EF(10.22)

where

M ≡ 1

N

∑

n

< Szn > (10.23)

and < Szn > is the expectation value of the d state spin polarization at site n.

It is convenient to define the parameter

δn ≡ n↑ − n↓ (10.24)

which can be expressed as:

δn =3nJ(0)M

EF(10.25)

One can calculate the change in the kinetic energy to lowest order ( which turnsout to be second order) in this parameter:


δEkin =(δn)2

6nEF (10.26)

We can write the total change in energy due to the interaction as:

∆Etot = −3

2

(J(0)M)2

EF(10.27)

Exercise 10.6Express the change in kinetic energy and the first order perturbation ∆E(1) inEq. (10.12) in terms of δn, and verify that the value (10.25) for δn is obtainedupon minimizing the total energy change with respect to δn.

We just performed averages over the whole system, so that we can considerthe equations above to be the result of a uniform mean-field treatment. How-ever we can also ask ourselves about the local non-uniform spin polarization ofthe conduction electrons, from the first order change in the wave functions ofthe conduction electrons.

We shall write the correction to each conduction electron ket -whose expres-sion contains local spin operators - due to the perturbation Hamiltonian (10.10).The conduction electrons are treated as non-interacting particles. Then we writethe wave function to first order in the perturbation expansion as

| kσ〉 =| kσ〉 +∑

k′σ′

(Hsd)k′σ′,kσ

εk − εk′

| k′σ′〉 (10.28)


Substitution of (10.10) into (10.28) leads to

| kσ〉 = | kσ〉 −∑

k′

J(k − k′)

εk − εk′

× 1

N

∑

n

ei(k−k′)·Rn (σSzn | k′σ〉 + Sσ

n | k′,−σ〉) (10.29)

where as usual the unperturbed Bloch functions were substituted by planewaves.

We can now obtain the spin-up and spin-down electron density of the con-duction band at low T , as

ρσ(r) =

kF σ∑

k

< r| k, σ > < k, σ |r > (10.30)

Since the local and conduction electron spins act independently as operators onthe corresponding kets, only diagonal terms in spin will be left upon calculatingthe average on the spin states, so we retain only these:

ρσ(r) =1

V

kF σ∑

k

( 1 − σ2m

h2N

∑

k 6=k′

J(k − k′)

k2 − (k′)2

×∑

n

ei(k−k′)·(r−Rn) + e−i(k−k′)·(r−Rn)〈Szn〉 ) (10.31)

With the aid of (10.22) one can rewrite the first term of (10.31) as

1

V

kF σ∑

k

=1

V

[

n+ 3σJ(0)

∑

n〈Szn〉

2EFN

]

(10.32)

Now call k − k′ = q, and perform the sum over k in the second term of (10.31).Let us define

f(q) =k2

F

N

∑

k≤kF

1

(k + q)2 − k2(10.33)

where we neglect the spin dependence of kF,σ , since it introduces a second ordercorrection. The second term of (10.31) then becomes

σ2m

h2V N

∑

q 6=0

J(q)f(q)∑

n

eiq·(r−Rn) + e−iq·(r−Rn)

〈Szn〉 (10.34)

By use of (10.20) we can write then the total densities as

ρσ(r) =1

V

(

n+ 3nσJ(0)

∑

n〈Szn〉

2EFN

)

+σ3n

8EFV N

∑

q 6=0

J(q)f(q)

×∑

n

ei(k−k′)·(r−Rn) + e−i(k−k′)·(r−Rn)〈Szn〉 (10.35)


We rewrite (10.33) as

f(q) =V

8π3N

∫

k≤kF σ

d3k1

q2 + 2qk cos θ(10.36)

which yields the result:

f(q) = 1 +4k2

F − q2

4qkFln

∣

∣

∣

∣

2kF + q

2kF − q

∣

∣

∣

∣

(10.37)

Exercise 10.7

• Prove (10.37);

• Show that limq→0 f(q) = 2 ;

• Show that ∂f(q)∂q is infinite at q = 2kF .

We see that the value of the second term in (10.35) is exactly the same of thethird term for q = 0, so one can just omit the second one and include q = 0 inthe third term.The final result is therefore

ρσ(r) =n

V+ σ

3n

8EFV N

∑

q

J(q)f(q)

×∑

n

eiq·(r−Rn) + e−iq·(r−Rn)Szn (10.38)

We note at this point that f(q) is, apart from a constant factor, the free electronsusceptibility for ω = 0, as can be verified by comparing it with Eqs. (7.76) to(7.78). If we take for simplicity J(q) ≈ constant ' J0, the spin polarization ofthe electron gas at r due to a non-zero average core spin at point Rn is just pro-portional to the space-dependent free-electron static susceptibility χ0(r −Rn),which is also the result of linear response theory in the static limit.


χ0(r −Rn) =1

N

∑

q

e−iq·R f(q) = −24πn

NΦ(2kFR) (10.39)

where

Φ(x) =x cosx− sinx

x4(10.40)

is Ruderman-Kittel range function.[4]Hint: you can use the integral representation:

ln

∣

∣

∣

∣

a+ q

a− q

∣

∣

∣

∣

= 2

∫ ∞

0

sin (ax) sin (qx)

xdx (10.41)

10.3. INDIRECT EXCHANGE HAMILTONIAN 247

One obtains finally

ρσ(r) =n

V− 2π(3n2)σJ0

NEF×∑

n

Φ(2kF | r −Rn |)〈Szn〉 (10.42)

The conclusion is that the perturbation of the spin-dependent density at thepoint r is a sum of contributions from all magnetic sites, each of which oscillatesin space with period 1/(2kF ) and decrease with distance as

|r −Rn|−3.

Yosida [5] observes that if the approximation J(q) = const. is improved byintroducing a cutoff in J(q) then the linear perturbation of the electron densityis finite at the position of the ions. To follow Yosida’s argument we remindourselves that in Exercise 10.7 we found that f(q) decreases abruptly at q = 2kF ,where the derivative is singular. We notice as well that it vanishes for q 2kF

as ∼ (q/2kF )−2. Since J(k,k + q) certainly vanishes as q → ∞ he makes thefollowing reasonable approximation:

f(q)J(q) =

2J0 if q ≤ 2kF

0 if q > 2kF(10.43)

Then the r dependent term in the r. h. s. of (10.38) is

−4σ(3n)2

V EF

J0

N

∑

n

2kF |r −Rn|Φ(2kF |r−Rn|)Szn (10.44)

which is finite at r → Rn.


10.3 Indirect exchange Hamiltonian

We shall now obtain the second order correction to the energy of the Fermi gasof conduction electrons due to the exchange interaction with localized spins.In the following we shall consider the localized-electron spins as fixed externalvariables, and we shall obtain an expression for the perturbation of the totalenergy of the system that will depend on the detailed configuration of the lo-calized spins. The result in the simplest possible form is an effective interactionamong the localized spins of the form of the Heisenberg isotropic Hamiltonian.Let us write the expression for the second-order correction to the energy of theelectron gas:

∆E(2) = 〈0∣

∣

∣

∣

HsdP

E0 −H0Hsd

∣

∣

∣

∣

0〉 (10.45)

where the projection operator P = 1 − |0〉〈0| eliminates the ground state ofthe Fermi gas from the sum over intermediate states. Upon substituting the


definition of Hsd from (10.10), we obtain the isotropic Heisenberg interaction,with an effective exchange integral of the form [7]:

Heff = ∆E(2) =∑

n,m

(J0)2

(

3n

N

)22π

EFΦ(Rn −Rm)Sn · Sm (10.46)

Exercise 10.10Verify Eq. (10.46) and show that in the case of free electrons and within the ap-proximation J(q) = constant = J0, Φ(x) is the Ruderman-Kittel range functionof Eq. (10.40).

We remark that the particular form (10.46) is obtained in the case of free elec-trons in a box and under the simplifying assumptions on J(q) that [8]:1) J(k,k′) ≈ J(k − k′);2) J(q) ≈ J0 .

We shall see in the following section that if the periodic lattice potentialis considered, the resulting band structure, and as a consequence, the usuallycomplex geometry of the Fermi surface, demand a non-trivial generalization ofthe RKKY formulation.

10.4 Range function and band structure

Let us now extend the theory by including more than one conduction band, andalso by representing the one electron band states by Bloch waves. Instead of(8.6) then we should write:

Jn n′(k,k′) = e2∫ ∫

dr1 dr2φ(r1)φ(r2)1

| r1 − r2 | ·

u∗kn(r1)uk′n′(r2)e−ik·r1eik′·r2 (10.47)

Then the range function in the general case is the sum

Φ(R) =∑

n,n′

Φnn′(R) (10.48)

over all relevant bands of the contributions due to each pair n, n′ (where n = n′

is the previous case and will be naturally included in the sum):

Φnn′(R) = − 1

2V 2

∑

kk′

In,n′(kk′)f(εkn) − f(εk′n′)

εkn − εk′n′

e−i(k−k′)·R (10.49)

whereIn,n′(kk′) =| Jn n′(k,k′) |2 (10.50)

Let us assume, as before, that I is slowly varying in k and k′. The sums over thereciprocal space in (10.49) are to be performed within the first BZ. By definition,the function In,n′(kk′) is invariant under time-reversal:

In,n′(−k − k′) = In,n′(kk′) (10.51)

10.4. RANGE FUNCTION AND BAND STRUCTURE 249

Figure 10.1: Orthogonal coordinates for integration in k space

We perform as usual the thermodynamic limit V → ∞ and substitute sums byintegrals over the first BZ.

To start with, let us define a system of coordinates in k-space appropriate todescribe arbitrary surfaces of constant energy. We choose for convenience thekz axis paralel to the vector R separating the two interacting localized spins.For the n-th band we perform the transformation

(kx, ky, kz) =⇒ (εkn, kl, kφ) (10.52)

from the original cartesian coordinates to a new system of orthogonal coordi-nates which are defined in the following way: [9, 10]

Let us consider the family of surfaces εkn = constant. The intersection ofthe surface of constant energy with the plane kz = const. defines a curve onthe plane. At any point P on such a curve we can define an orthogonal basistriad, with versors along, respectively, the gradient of the energy ∇kεkn (whichis perpendicular to the surface), the tangent to the curve and the tangent tothe constant energy surface which is perpendicular to both previous versors, asshown in Fig. 10.1. We call the differentials of k along those directions dkε, dkφ

and dkl respectively.


The volume differential in k space in a neighbourhood of P is:

dV = dkφdkz

(

∂kl

∂kz

)

ε,kφ

dε

| ∇ε | (10.53)

Let us choose the sense of kl such that(

∂kl

∂kz

)

ε,kφ

> 0. The double volume

integral in k space of (10.49) can now be performed in the new coordinatesystem. We perform first the curvilinear integrals for given energies ε and ε′

and fixed kz, k′z which yield as a result a function γ of all these four variables:

γ(ε′n, k′z ; εn, kz) =

1

(2π)6

∮

ε, kz

∮

ε′, k′z

dkφ

| ∇kεn(k) |

(

∂kl

∂kz

)

×dk′φ

| ∇k′ε′n(k′) |

(

∂k′l∂k′z

)

Inn′(k,k′) (10.54)

We assume that Inn′(k,k′) and ∇kεn(k) are slowly varying functions of kφ, sothat we can perform the curvilinear integral over kφ and use

dS = dkl

∮

dkφ (10.55)

as element of surface area on the constant energy surface. Then we can rewriteEq. (10.54) as the second derivative of a double surface integral

γ(εn′ , k′z ; εn, kz) =1

(2π)6∂

∂kz

∂

∂k′z

∮

ε, kz

∮

ε′, k′z

× dSn

| ∇kεn(k) |dSn′

| ∇k′εn′(k′) |Inn′(k,k′) (10.56)

Notice that integrations in (10.56) are performed over the curves defined byεn = const., kz = const., and the corresponding conditions for εn′ and k′z.

Exercise 10.11Prove the equivalence of (10.54) and (10.56) under the present assumptions.

Finally, the expression for Φnn′ is obtained by integrating with respect to kz,k′z and the energy variables:


Φnn′ = −1

2

∫ ∫ ∫ ∫

dεndεn′dkzdk′zγ(εn′ , k′z ; εn, kz)

×[

f(εn) − f(εn′)

εn − εn′

]

eiz(kz−k′z) (10.57)

Consider a given constant energy surface and assume, for the time being, thatit is contained within a finite volume inside the first BZ. Then in the simplestcase of a simply connected, smooth surface, of which a sphere is the ideal case,one finds several extreme values of kz . Since the energy eigenvalues are real andinvariant under inversion in k space, for each extreme value of kz, say ki, thereis another one kj = −ki. Let us now label all extreme values of kz for bothenergy surfaces, as ki, k

′j . We sketch a simple example in Fig. 10.2.

Let us change notation, and, since z was chosen in the direction of R, call

Figure 10.2: Extreme values of kz on constant energy surfaces in k space

z = R. The expression (10.57) is particularly adequate for obtaining an asymp-totic expansion of the range function for large R. To this end it is convenientto use the identity

eikR ≡ 1

iR

∂

∂k(eikR) (10.58)


and rewrite (10.57) as:

Φnn′ =1

2

∫ ∫ ∫ ∫

dεnd εn′dkzdk′zγ(εn′ , k′z ; εn, kz)

×[

f(εn) − f(ε′n)

εn − εn′

]

∂

∂kz

∂

∂k′z

(

eiR(kz−k′z))

(10.59)

The double integral over kz , k′z above can be performed by parts. The end

points of the interval for the kz and k′z integrations are the extreme values. Onecan use repeatedly transformation (10.58) and integration by parts to obtain anexpansion of Φ in inverse powers of R, which one expects will converge for largeR. We write it down below, simplifying a bit the notation, by using k, k′ for kz

etc. and defining∆f

∆ε=f(εn) − f(ε′n)

εn − εn′

. (10.60)

We find

Φnn′ = − 1

2R2

∑

j,l

∫ ∫

dεndεn′Γj,l( ; εn, εn′ , R)

× ∆f

∆εeiR(kj−k′l )ei(φj−φl) (10.61)

where

Γj,l(ε, ε′, R) = γ(ε, kj ; ε

′, k′l) −1

iR

(

∂γ

∂k− ∂γ

∂k′

)

kj ,k′l

+1

(iR)2

(

∂2γ

∂k2− 2

∂2γ

∂k∂k′+∂2γ

∂k′2

)

kj ,k′l

+ · · · (10.62)

The phase factors eiφi , eiφj from upper/lower limits are respectively +1 for amaximum, −1 for a minimum.

For large enough R we can neglect all but the first term in (10.62). In thiscase, Γ reduces to

Γj,l(ε, ε′, R) ' Inn′(kj,k

′l)

(2π)6 | ∇kjεn || ∇kl

εn′ |

×(

∂Sn

∂kz

)

kj

(

∂Sn′

∂kz′

)

k′l

(10.63)

In the cases of ellipsoidal or spherical surfaces, γ is independent on kz and k′z,so that (10.63) is in fact exact [10]. The special cases of a sphere and an el-lipsoid with axial symmetry around the direction of R can be easily worked out.


Exercise 10.12Prove that if I is constant and if the constant-energy surface is an ellipsoid ofrevolution with symmetry axis parallel to R, so that

ε =h2

2

[

(k2x + k2

y)

m∗‖

+k2

z

m∗⊥

]

(10.64)

one gets

Γ =4π2I

h2 m∗⊥m

∗′⊥ (10.65)

The case of saddle points can be analyzed by similar methods. Following Rothet al. [10] we can write an unique expression for all kinds of extreme points,namely bulges, dimples and saddle points, as

Φ(R)nn′ = −∑

jl

ei(φj−φl)

2R2

∫ ∫

dεndεn′ΓjleiR(kj

z−klz) f(εn) − f(εn′)

εn − εn′

(10.66)

Here the expression for Γ has the general form

Γjl(R, εn, εn′) = γjl(εn, εn′) −γ

(1)jl (εn, εn′)

iR

+γ

(2)jl (εn, εn′)

(iR)2+ · · · (10.67)

When the surface of constant energy has bulges, dimples and saddle points, asin the one shown in Fig. 10.2, the integral by parts we must perform to obtainγ has to be broken into the contributions from the different extrema, and thesum over ki, k

′j contains all these separate contributions [10]. The last factor

in (10.66) involves the Fermi functions in an expression which has a maximumwhen both band energies are near the Fermi level, so let us assume that most ofthe contribution to the double integral comes from that region. Then we extendthe energy integrals to ±∞. This approximation is quite reasonable even athigh temperatures, because the bandwidths are typically several eV , while theenergy interval where the Fermi function varies appreciably is of the order ofkBT around the Fermi energy EF . The integrals can be performed by closingthe contours in the complex plane. This requires the analytic continuation ofkj and kl as functions of ε or ε′ to complex values of the energies. In practice,though formally extending the integrals to infinity, one only needs the propertiesof these functions near the real axis. Let us assume that k(ε) is analytic, so thatwe expand it in a series around the real value εr

n for complex εn = εrn + iεi

n:

kjz(εn) = kj

z(εrn) +

(

dkjz(εn)

dεn)

)

εrn

· iεin + · · · (10.68)


The sign of the z component of the electron group velocity

vz =

(

dεn

dkjz

)

εrn

at the extremum, determines whether the contour must be closed on the upper orlower half- plane. Analytic continuation of the dispersion relation of elementaryexcitations is a standard problem in solid state physics, as in the calculation ofsurface or interface states [11], the localization properties of Wannier functions[12], etc. Let us consider the integral over εn. The function

g(εn) =f(εn) − f(εn′)

εn − εn′

(10.69)

is regular, at finite temperature, for εn → εn′ . We can close the contour in thehalf plane in which the integrand converges exponentially at infinity, that is, forthe ε integral, that where

vnjz · Im(εn) > 0 .

The poles which contribute with finite residues to the Cauchy integral are εn =EF + iωmj

, where, if n is the band index , j denotes one particular extremumof that band, and σnj = sign(vnj

z ), we have

ωmj= (2mj + 1)πkBTσnj , (10.70)

Therefore, calling β = 1/kBT , we obtain the following expression for (10.66):

Φ(R)nn′ = − 2πi

2R2β

∑

lj

∞∑

mj=0

σnjei(φj−φl) ×

∮

dεn′e−iRk′lz

EF + iωmj− εn′

[

Γjl(R, εn, εn′)eiRkjz

]

εn=EF +iωmj

(10.71)

Due to the minus sign of the exponent in the factor e−iRk′lz the integral over

εn′ converges on a contour for which vzv′z < 0 Therefore, there will be no

contribution from residues to the integral unless the condition

∂kjz

∂ε· ∂k

′lz

∂ε′< 0 (10.72)

is satisfied. The conclusion is that a given pair (j, l) of extrema in k space con-tributes to the range function only if the z components of their group velocitieshave opposite sign.

The resultant expression for the range function is:

Φ(R))nn′ =2πi

2R2β

∑

jl

′∞∑

m=0[

ei(φj−φl)eiR(kjz−k′l

z )Γjl(R, εn, εn′)]

εn=εn′=EF +iωmj

(10.73)


where the prime in the summation means that it is limited to the case vjz ·vl

z < 0.We expand now the term in square brackets in (10.73) around EF , assumingthat Γ can be approximated by a constant. The expansion of the exponentyields:

(

kjz − kl

z

)

εn=εn′=EF +iωmj=

(

kjz − k′lz

)

+ iωmj

(

∂kzj

∂εn− ∂k′lz∂εn′

)

EF

+1

2(iωmj)

2

(

∂2kjz

∂ε2n− ∂2k′lz∂ε2n′

)

EF

+ · · · (10.74)

We assume that terms involving the second derivatives of kz above are small,and neglect higher order derivatives. Note that the pairs kj, − kj as well asφj ,−φj etc. appear always associated in the sum over j, l in (10.72), so thatthe result of (10.73) is real. Pairs of points like (kj,−kj) are called calipersbecause 2 | kz

j | is a diameter of the Fermi surface in the direction of R. Afterperforming the summation over m in (10.73), we arrive at [9]:

Φn,n′ =∑

j,l

′In,n′ (kj(εn), kl(εn′)EF

)m∗j (EF )m∗

l (EF )

16π3h4R3

×(

cosψ(j, l) +bjl

Rsinψ(j, l)

∂2

∂a2jl

)

πRkBT

sinh (ajlπRkBT )(10.75)

where

ψ(j, i) ≡ R · (kj(εnt)EF− kl(εn′)EF

+ φj − φl (10.76)

ajl ≡∣

∣

∣

∣

∣

(

∂kjz(εn)

∂εn

)

EF

−(

∂klz(εn′)

∂εn′

)

EF

∣

∣

∣

∣

∣

(10.77)

and

bjl ≡1

2

∣

∣

∣

∣

∣

(

∂2kjz(εn)

∂ε2n

)

EF

−(

∂2klz(εn′)

∂ε2n′

)

EF

∣

∣

∣

∣

∣

(10.78)

The summation in (10.75) extends over all calipering pairs of points on the Fermisurface, and also on spin.


The temperature dependent term on the r.h.s. of (10.75) yields an exponen-tially decreasing amplitude at large R. At low temperatures the range cutoff

Rc =hvz

F

kBT


can be very much larger than the mean free path due to electron scattering offimpurities or defects [1]. In concrete cases however the geometry of the Fermisurface can produce small values of the z component of the Fermi velocity atsome particular extrema, yielding shorter cutoff lengths.

For certain directions of R there might be no calipering points contributingto (10.75). This happens in particular when along that direction the Fermisurface touches the boundary of the Brillouin zone. In this case, due to thesimultaneous effect of time reversal and translation symmetry, the constantenergy surface must be orthogonal to the boundary and the present method ofevaluation of the range function does not apply [10].

We remark that the range function is in general multi-periodic in R, withwavelengths determined by the various calipers of the Fermi surface for thatdirection.

We can verify that Eq. (10.75) reduces to the simple RKKY result at low Tfor a spherical FS. In this case

ε(k) =h2k2

2m∗.

In the T → 0 limit we find the original RKKY expression:

Φ(R) = −m∗k4

F I(kF,−kF)

π3h2

(

x cosx− sinx

x4

)

where we included a factor 2 for spin and a second factor 2 due to the fact thatthe two equivalent caliper pairs (kF ,−kF ), (−kF ,kF ) must be considered.

Exercise 10.14Prove the above result and verify that the higher order terms in powers of 1/Rexactly cancel each other in the limit T → 0.

10.5 Semiconductors

There are several different cases to consider, depending on the temperature andthe doping level. In the case of intrinsic semiconductors, that is those witha negligible concentration of electronically active impurities, we must considerseparately the limits of high and low temperature. Here, ”high” or ”low’ refersto the value of the ratio kBT/Eg where Eg is the gap at T = 0 between theuppermost occupied (valence) band and the lowest unoccupied (conduction)band.

10.5.1 Intrinsic semiconductors, high T

At T > 0 there is a finite concentration of electrons in the conduction band andholes in the valence band. The carriers, within the effective mass approximation,can be described as two Maxwell-Boltzmann gases, one of electrons and one of

10.5. SEMICONDUCTORS 257

holes. We simplify the problem assuming that the dominant contribution is dueto the conduction band. This would be the case if m∗

e m∗h. Let us then

calculate the contribution to Φ(R) from a spherical band with effective massm∗ at high T . The high T form of the Fermi-Dirac distribution for electrons inthe conduction band is [13]

f(ε−Ec) 'nh3

2(2πm∗kBT )3/2e−(ε−Ec)/kBT (10.79)

where Ec is the bottom of the conduction band. We can apply equation (10.66).Let us define the new variables:

kR = x , k′R = y , b2 = h2/(2m∗kBT ). The exponential Boltzmann factorallows us to extend the integral over energy in the conduction band up to infinity.Then (10.66) becomes

Φ(R) = − nhI

4π4(2πm∗kBT )3/2×

∫ ∞

−∞

∫ ∞

−∞

(

e−b2x2 − e−b2y2)

x2 − y2ei(x−y)xy dx dy (10.80)

The integral in (10.80) may be performed by contour integration followed bythe use of the definite integral

∫ ∞

0

e−b2t2 t sin t dt =

√π

2b3e−1/b2 (10.81)

Exercise 10.15Prove that the final result is

Φ(R) =nm∗I

8πh2Re−(R/R0)

2

(10.82)

where the screening length R0 is

R0 =

√

(

h2

2m∗kBT

)

. (10.83)

10.5.2 Intrinsic semiconductors, low T

In this case the indirect interactions are due to electron excitation from a filledspherical valence band of effective mass m∗

h to an empty spherical conductionband with effective mass m∗

e , separated by the energy gap Eg . We start from(10.66), assume as usual that I is independent on k,k′, and define new variables

x = ke

√

h2

2m∗eEg

, y = ke

√

h2

2m∗hEg


and the parameters

a1 = R

√

2m∗eEg

h2 , a2 = R

√

2m∗hEg

h2 .

Now the expression for Φ becomes:

Φ = −C∫ ∞

−∞

∫ ∞

−∞

ei(a1x−a2y)

1 + x2 + y2xy dx dy (10.84)

where

C =2IEgm

∗em

∗h

h4R2

Exercise 10.16Verify (10.84).

We perform an orthogonal coordinate transformation in the plane x, y, suchthat the new axis x′ is in the direction of a1x− a2y, by defining:

(a21 + a2

2)1/2x′ = a1x− a2y

(a21 + a2

2)1/2y′ = a1x+ a2y (10.85)

Now one can perform the integral over x′ by contour integration and obtain:

Φ =4πa1a2C

η2

∫ ∞

0

e−η√

1+y′2

√

1 + y′2· (1 + 2y′2) dy′ (10.86)

where η = (a21 + a2

2)1/2 is proportional to R, so that in the large R limit we can

evaluate the integral approximately, obtaining:

Φ = −2IE2g

h6

(m∗em

∗h)3/2

2π5/2

1

(k0R)5/2e−k0R (10.87)

where the inverse screening length is

k0 =

√

2Eg(m∗e +m∗

h)

h2 .


Semiconductors are therefore capable of sustaining indirect exchange interac-tions among magnetic impurities. However, in distinction with the case of met-als, the results above predict an effective interaction which does not oscillate in

10.6. MAGNETIC MULTILAYER SYSTEMS 259

space. Detailed calculations by Narita [14] -which also include the case of metal-lic systems with a saddle point Fermi surface- generalize previous calculations[15] and show that the sign of the interaction depends on the band parameters.Experimental results for systems called magnetic ”trilayers” (see next section),[17] in which two ferromagnetic layers are deposited on a convenient substratewith a non-magnetic layer in between, called ”spacer”, reveal an effective cou-pling among the magnetic layers whose sign depends upon the spacer width,even in cases where the latter is a semiconductor. Within the simple modeldiscussed above this is not possible when the Fermi level in the semiconductorlies inside the gap.

10.5.3 Degenerate semiconductors

For very highly doped semiconductors at low temperatures one may find theFermi level inside a partially occupied band and, at low enough temperaturesthe semiconductor transforms into a low conductivity metal with either kind ofcarriers. This happens in covalent semiconductors, typically Ge and Si, whenthe doping impurity concentration is large enough that a reasonable statisticalfraction of their wave fuctions start to overlap and generate a band of stateswith energies contained in the gap [18]. At low enough temperatures ( in generalabout a couple of K) the electrons (holes) for n(p) type systems become adegenerate Fermi liquid.These systems are completely disordered, beause theimpurities occupy random substitutional sites in the host lattice. Therefore, theelectronic impurity states are solutions of Schrodinger’s equation with a randompotential, and the absence of translational symmetry precludes the applicationof Bloch’s theorem. The k vector is no longer a ”good quantum number” andaccordingly the concept of Fermi surface simply does not occur in this context.Neglecting Coulomb interactions between the carriers it is possible to obtain atheoretical description of the impurity bands which leads to a range function forthe indirect exchange interaction among eventual diluted magnetic impuritiesthat oscillates in space [19], in spite of the absence of a Fermi surface. One alsofinds an oscillating range funcion in a metallic disordered alloy [20, 21].

Finally, let us mention that in mixed finite clusters of a transition metalsurrounded by noble metal atoms, long-range magnetic polarization oscillationsare also found, both theoretically [27] and experimentally [28].

10.6 Magnetic multilayer systems

These are systems produced artificially in the laboratory in which ferromag-netic and non-magnetic layers alternate along the direction perpendicular tothe layer planes. They have been the object of intense research since the obser-vation by Grunberg et al. [17] that Fe films separated by a Cr spacer coupledantiferromagnetically. Upon varying the width of the spacer, Parkin [22] et al.discovered that the interlayer effective interaction in Fe/Cr/Fe and Co/Ru/Comultilayers changes alternatively from FM to AFM when the spacer thickness


varies. The layers, either of magnetic or non-magnetic elements, can containfrom just a few to a hundred atomic layers. The oscillations are found withany non-magnetic transition metal as spacer. The oscillations, which in generalare multi-periodic, are in many cases consistent with the generalized RKKYtheory presented above, and in principle depend mainly on the properties ofthe band structure of the spacer metal. Further study has shown that thereis an alternative theory which describes these oscillations as originating in theconfining effect of the spin-dependent potential barriers at the interface regionbetween both materials. In the FM alignment in which all majority-bands arebelow the Fermi surface (complete ferromagnetism), electrons of this spin withenergy EF are confined to the non-magnetic spacer. This is accordingly knownas the “quantum-well” theory of interlayer coupling [23]. It has been found thatit is possible to obtain both theories as limiting cases of a general approach inwhich a total energy calculation of a multilayer is performed based on Green’sfunctions methods [24]. We refer the interested reader to some excellent reviewarticles [25, 26].

References

1. De Gennes, P. G. (1962) J. Phys. Radium 23, 510; 630.

2. Elliott, R. J. (1965) Magnetism, eds. Rado, G. T. and Suhl, H. H. 2a,Academic Press, New York, p. 355.

3. Kittel, C. (1971) “ Introduction to Solid State Physics”, Chap. 15, JohnWiley & Sons, Inc., New York.

4. Ruderman, M. A. and Kittel, C. (1954) Phys. Rev. 96, 99.

5. Yosida, K. (1957) Phys. Rev. 106, 893.

6. Specht, Frederick (1967) Phys. Rev. 162, 389.

7. Kasuya, T. (1956) Prog. Theoret. Phys. 16), 45.

8. Kaplan, T. A. (1961) Phys. Rev. 124, 329.

9. Zeiger, H. J. and Pratt, G. W. (1973) “Magnetic interactions in solids”,Clarendon, Oxford.

10. Roth, Laura M., Zeiger, H. J. and Kaplan, T. A. (1966) Phys. Rev. 149,519.

11. Chaves, C. M., Majlis, N. and Cardona, M. (1966) Sol. State. Comm. 4,631.

12. Blount, E. I. (1962) “Solid State Physics, Adv. in Res. and Appl.”eds. F. Seitz and D. Turnbull, Vol. 13, pp. 306-70.

10.6. MAGNETIC MULTILAYER SYSTEMS 261

13. Ziman, J. M., (1964) “Principles of the Theory of Solids”, Cambridge atthe University Press.

14. Narita, Akira (1986) J. Phys. C19, 4797.

15. Baltensperger, W. and de Graaf, A. M. (1960) Helv. Phys. Acta 33,881.

16. Bloembergen, N. and Rowland, T. J. (1955) Phys. Rev. 97, 1679.

17. Grunberg, P., schreiber, R., Pang, Y., Brodsky, M. B. and Sower, H.,(1986) Phys. Rev. Lett. 57, 2442.

18. Majlis, N. (1967) Proc. Phys. Soc. London 90, 811.

19. Ochi, Carmen L. C.. and Majlis, N. (1995) Phys. Rev. B51, 14221.

20. Bulaevskii, L. N. and Pannyukov, S. V. (1986) JETP Lett. 43, 190.

21. Bergmann, Gerd (1987) Phys. Rev. B36, 2649.

22. Parkin, S. S. P., More, N. and Roche, K. P. (1990) Phys. Rev. Lett. 64,2304.

23. Edwards, D. M., Mathon, J., Phan, M. S. and Muniz, R. B. (1991) Phys.Rev. Lett. 67, 493.

24. d’ Albuquerque e Castro, J., Ferreira, M. S. and Muniz, R. B. (1994) Phys.Rev. B 49, 16062.

25. Bruno, P. (1995) Phys. Rev. B52, 411.

26. Fert, A. and Bruno, P. (1992) “Ultrathin Magnetic Structures”,eds. Heinrich, B. and Bland, A., Springer–Verlag.

27. Guevara, Javier, Llois, Ana Maria and Weissmann, Mariana (1998) Phys.Rev. Lett. 81, 5306.

28. Guevara, Javier and Eastham, D. A. (1997) J. Phys.-Condens. Matter9, L497.

Chapter 11

Local Moments

11.1 The s-d and Anderson Hamiltonians

In this chapter we shall study the process of formation of local magnetic mo-ments and the effect of substitutional magnetic impurities in a host non-magneticmetal.

The basic models which have been the basis of the theoretical study of theseproblems are the s-d and the Anderson model. The s-d exchange interaction wasalready introduced in Chap. 10, where we derived the form of the long-rangeRKKY interaction among spins localized on different sites in a non-magneticmetal, like Mn in Cu.

However, when iron-group elements are introduced into a non-magneticmetal, it is not always the case that they display a permanent magnetic mo-ment. Fe and Mn in Cu maintain their spins, while Mn in Al does not. Forthe first case, we may use the s-d exchange interaction model of Chap. 10. Thisinteraction is the main mechanism of the Kondo effect, that we shall discussfurther on in the second part of this chapter.

In the second case, one must understand why there is no magnetic momenton the transition ion impurity. In order to study the conditions for an impurityto sustain a permanent magnetic moment we require an approach departingfrom a more basic standpoint, such that both cases can be explained with thesame model.

11.2 Anderson model

To this end Anderson proposed the following Hamiltonian:[1]

H =∑

kσ

εka†kσakσ +Ed

∑

σ

a†d σad σ

263

264 CHAPTER 11. LOCAL MOMENTS

+∑

kσ

Vk da†kσad σ + Vd ka

†d σakσ + U a†d ↑ad ↑a

†d ↓ad ↓ (11.1)

where Ed, assumed to lie below EF , is the energy level of the electron boundto the transition metal ion whose location one may choose as the origin of thecoordinate system. The conduction electrons are taken as mutually independent,while Hubbard repulsion acts on the impurity bound electrons. Degeneracy ofthe d states is also neglected. The parameters above are:

U =

∫

d r1d r2 |φd(r1)|2 |φd(r2)|2e2

r12(11.2)

Vk d =1√N

∫

d3 r∑

n

φd(r)Vion(r)Wk(r −Rn)eik·Rn (11.3)

with Wk(r −Rn) = Wannier function, centered at site n, for a conductionelectron.

11.3 Hartree–Fock solution of Anderson Hamil-tonian

We can define four retarded Green’s functions for this problem:

Gσk k′ ≡ 〈〈akσ(t) | a†k′σ(0)〉〉

Gσk d ≡ 〈〈akσ(t) | a†d(0)〉〉

Gσd d ≡ 〈〈adσ(t) | a†dσ(0)〉〉

Gσd k ≡ 〈〈adσ(t) | a†kσ(0)〉〉 (11.4)

Observe that Gk,k′ is not diagonal in the quasi-momentum variables, becausethe presence of an impurity at a given site breaks the translation symmetry ofthe Hamiltonian.

We repeat now for the Hamiltonian defined in Eq. (11.1) the procedure fol-lowed in Chapters 6 and 9 to calculate the retarded Green’s functions:

we write down the equations of motion for all Green’s functions, performthe time Fourier transform to the frequency domain, and whenever a product of4-fermion operators appears we substitute in all possible ways binary productsof a creation times an annihilation operator by their statistical average, to becalculated self-consistently afterwards.

Exercise 11.1Verify that the result is the following system of algebraic linear equations:

(ω −Eσ)Gσd d −

∑

k

Vd k Gσk d = 1

(ω − εk)Gσk,d − Vk dG

σd d = 1

11.3. HARTREE–FOCK SOLUTION OF ANDERSON HAMILTONIAN 265

(ω −Eσ)Gσd k −

∑

k′

Vd k′ Gσd = 1

(ω −Eσ)Gσk k′ − Vk dG

σd k′ = 1 (11.5)

where we definedEσ = Ed + U〈nd−σ〉 (11.6)

We introduce now the auxiliary function

Sk(ω) =∑

k′′

Vd k′′ Gk′′ k(ω + iη) (11.7)

and we find:

Sk(ω) =Vd k gk(ω)

1 − gd(ω)∑

k′′ gk′′(ω)Vk′′ d(11.8)

where the unperturbed Green’s functions are

gk(ω) =1

ω − εk

gd(ω) =1

ω −Ed(11.9)

Finally, we obtain:

Gσd d = (ω −Eσ − Σ(ω + iη))

−1

Gσk k′ (ω) = gk(ω) δk k′ +

gk(ω)Vk d Vd k′ gk′(ω)

1 − gd(ω) Σ(ω)(11.10)

where Σ(ω), the local state self-energy, is

Σ(ω) =∑

k′′

Vd k′′gk′′(ω)Vk′′ d ≡ ∆(ω) + iΓ(ω) (11.11)

The imaginary part of Σ is

Γ(ω) ≡∑

k

|Vd k|2ω − εk + iη

= −iπ∫

dεk

∮

εk=ω

dSk |Vd k|2|∇k εk|

≡ ρ(εk)V 2(εk) (11.12)

where V 2(εk) is the average of |Vd k|2 over the surface εk = ω and ρ(εk) is theunperturbed density of states of the conduction electrons.The real part of the local self-energy is

∆(ω) = P

∫

d εkρ(εk)V 2(εk)

ω − εk(11.13)


Let us suppose that V 2(εk) is only appreciable in an interval of energies overwhich ρ(E) is almost constant. Then the principal value integral is very small,and we can neglect the displacement ∆ of the real part of the energy. As afurther approximation we take ρ(E) at the fermi energy. Now we get the localdensity of states as

ρd σ(ω) = − 1

πImGσ

d d =1

π

Γ

(ω −Eσ)2 + Γ2(11.14)

and we can immediately calculate the average number of d electrons at T = 0for a given spin, which enters into the definition (11.6 ) of the effective localstate binding energy Eσ . We obtain a pair of self-consistent equations to solve,for 〈nd ↑〉 and 〈nd ↓〉:

〈nd ↑〉 =

∫ EF

−∞

ρd σ(ω)dω =1

πcot−1

(

Ed + U〈nd↓〉 −EF

Γ

)

〈nd ↓〉 =

∫ EF

−∞

ρd σ(ω)dω =1

πcot−1

(

Ed + U〈nd↑〉 −EF

Γ

)

(11.15)

Clearly, they have always a non-magnetic solution

〈nd↑ 〉 = 〈nd ↓ 〉 = nd .

This is certainly the case for U/Γ = 0, because for this situation E↑ = E↓ andboth equations are identical. Therefore one suspects that magnetic solutionsonly exist when this ratio is greater than some minimum value. Let us simplifythe notation. We define n1 ≡ 〈nd ↑ 〉 , n2 ≡ 〈nd ↓ 〉. It is clear from the structureof the pair of equations (11.15) that the permutation of the spin orientationn1 ↔ n2 is a symmetry operation. In the plane (n1, n2) this operation is themirror reflection of the curves n2 = F (n1), n1 = F (n2) on the line n1 = n2. Inthe absence of a magnetic solution both curves coincide, and the common curvemust be normal to the line n1 = n2 at the point where they intersect, or , inother words, the tangent of the curve is a line n1 + n2 = const. In order thatthere be a magnetic solution within the physical domain, which is the square0 ≤ ni ≤ 1, both curves must not coincide, and therefore they must have at leastthree intersections: one, on the mirror line, one for n1 > n2 and a symmetricone for n2 > n1. This implies that at the intersection of both curves with themirror line, the derivative of, say, n1 = F (n2), must be > 1 in absolute value,which implies

|F ′(n2)|n1=n2 > 1 (11.16)

that isU

∆≥ π

sin2 πnd

(11.17)

where we substituted n2 = nd to obtain the limiting case in which the magneticsolution just appears. Comparing the r. h. s. of Eq. (11.14) with that of theinequality above and using Eq. (11.15), we can rewrite (11.17) as

ρd(EF )U ≥ 1 (11.18)

11.4. KONDO EFFECT 267

where ρd(EF ) is the density of states of d electrons at the Fermi level for thenon-magnetic solution with the same parameters. We see that this conditionhas a familiar form.

One suspects that the zero-field static susceptibility would diverge when theequality applies in (11.18). Let us verify this. Under an infinitesimal appliedmagnetic field δ B the Hartree–Fock energy of the local state changes by

δ Eσ = −g2σµB δ B + Uδ 〈nd−σ 〉 (11.19)

The local susceptibility of the d state is

χd = −g µB

2

(δ n↑ − δ n↓)

δ B(11.20)


χd =1

2

g2µ2Bρd(EF )

1 − ρd(EF )U(11.21)

which behaves as expected.

It is possible to approach the study of Anderson’s Hamiltonian with pertur-bation methods. The two cases in which this is reasonable are either the largeU , small limit, or the inverse one with large Vkd, small U . For the iron group Uis estimated to be large. In this limit, assuming (U,Ed) large compared to Vkd,second order perturbation in the exchange interaction leads to the s-d modelhamiltonian, which is the basis of the Kondo effect [2] The complete Hamilto-nian for this case is the sum of the band Hamiltonian with independent electronsand the exchange Hsd Hamiltonian of Eq. (10.10):

H =∑

k σ

εk σ ˆnk σ +Hsd (11.22)

which we used to study the case where there are permanent magnetic momentson substitutional impurities in the metal lattice.

11.4 Kondo effect

According to the results of the previous section, a non-zero magnetic momentrequires, within the Hartree–Fock approximation, that inequality (11.18) besatisfied. As mentioned before, this case leads to the Kondo effect, which wasdiscovered while looking for the explanation of the behaviour of the resistanceof an alloy like Cu-Mn as a function of temperature.

The electrical resistance R(T ) of normal metals at large T is dominated bythe electron-phonon interaction [3]. At high T > ΘD it grows linearly with T .For T ΘD it decreases as T 5. Therefore, as T → 0, the phonon contribution


to electron scattering becomes negligible. Due to the unavoidable presence ofimpurities which are responsible for elastic scattering of conduction electrons,

limT→0

R(T ) → constant× impurity concentration ≡ R0 ,

which is called the residual resistance.Experiments performed in the 30’s had already shown that the electrical

resistance R(T ) of some alloys of a noble metal with a small (< 0.1%) atomicconcentration of a magnetic element has a minimum at low T ≈ 10K. [4, 5]

The explanation of this minimum was found by Kondo in 1964.[6]. He cal-culated the scattering amplitude for conduction electrons by a localized spin Sdue to the exchange interaction of Eq. (10.10), and he discovered that in thesecond order of the Born perturbation series, the scattering amplitude displaysa singular behaviour which leads to a lnT low temperature dependence of theresistance. For an AF exchange (J < 0) this term increases as T decreases,while the phonon contribution decreases, as we have just mentioned, so thecompetition between these opposite behaviours explains the minimum of R(T ).At very low T the logarithmic term would however diverge as T → 0, whileit was found experimentally that the resistance deviates from its T = 0 valueby a term in T 2, so that Kondo’s calculation had to be modified. A detailedsurvey, though, of the evolution of the ”Kondo problem” is outside the scope ofthis book. I refer the interested reader to the excelent monograph by Hewson [7].

11.4.1 Calculation of resistivity

We consider the simple case in which the conduction band electrons are welldescribed as almost free, with kinetic energy

εk =h2k2

2m∗

where m∗=effective mass. In the following we shall not write the asterisc on theeffective mass to simplify the notation. In the steady state, under the applicationof a constant uniform electric field F ,the electron distribution function in fσ(k)satisfies the Boltzmann equation:[8]

(

∂fσ(k)

∂t

)

field

+

(

∂fσ(k)

∂t

)

scatt

= 0 (11.23)

The field time-derivative is(

∂fσ(k)

∂t

)

field

=∂fσ(k)

∂εk∇kεk · dk

d t(11.24)

The time-derivative of the quasi-momentum is:[9]

dk

d t= (e/h)F (11.25)


so that we have:(

∂fσ(k)

∂t

)

field

=∂fσ(k)

∂εk∇kεk · (e/h)F (11.26)

where e = electron charge.Let us now obtain the scattering term. Collisions with impurities are respon-

sible for a finite life-time of an electron state. We consider isotropic scattering,so that the life-time τc depends only on the electron energy. In this case, a de-viation from the equilibrium distribution f0(k) at the fixed temperature decaysaccording to

(

∂fσ(k)

∂t

)

scatt

= −f − f0τc

(11.27)

where τc(εk) is the collision time, and (τc(εk))−1 is the transition probabilityper unit time for the electron to be scattered from state k. We obtain thereforefrom Eq. (11.23):

f(k) − f0(k) = −∂f0(k)

∂εk∇kεk · (e/h)Fτc(εk) (11.28)

to first order in the field, and assuming that there is no spin dependence of thecollision time. We can now calculate the electric current density in a sample ofvolume V with the field applied in the x direction as

jx =2e

V

∑

k

hkx

m(f(k) − f0(k)) (11.29)

where the factor 2 is included to take care of both spins.

Exercise 11.3Show that the conductivity, defined as

σ =∂jx∂F

(11.30)

has the form

σ = −4e2

V

∫

d εkρ(εk)τc(εk) εk∂f0(εk)

∂εk(11.31)

We remark that the last factor in Eq. (11.31) imposes that |εk −EF | ≤ kB T .

11.4.2 Calculation of the collision time

In order to complete the calculation of σ we need to obtain the collision timeτc. We shall do this within the formal scattering theory approach, and we shallgo up to the second Born approximation.

We call Ho the unperturbed Hamiltonian of the conduction band electrons


in the host lattice potential and V the Hsd exchange interaction.We need to calculate the transition probability per unit time W (k, σ →

k′, σ′) from an incoming state a†k↑|0〉, where |0〉 is the sea of conduction elec-

trons, to an outgoing state a†k′↑|0〉 or a†k′↓|0〉 which is expressed through Fermi’s

”Golden Rule” in terms of the T matrix:

W (k, σ → k′, σ′) =2π

h

∑

M

wM |T (k′, σ′, M ′ |k, σM) | 2δ(εk − εk′) (11.32)

where wM is the probability that the magnetic impurity initial state be Sz = M .We remind at this point the definition of the T matrix.

The outgoing state, denoted |k〉+, satisfies the Schrodinger equation

|k〉+ = |k〉 +1

εk − H0 + iηV |k〉+ (11.33)

where the infinitesimal η defines the outgoing (retarded) behaviour in time. Onedefines the transition matrix T as

|k〉+ = T (εk + iη)|k〉 (11.34)

We verify that T can be formally expanded in the series

T (z) = V + V G0(z)V + V G0(z)V G0(z)V + · · · (11.35)

where the resolvent operator G0(z) is

G0(z) ≡1

z − H0

(11.36)

Due to the scalar character of the exchange interaction the transition matrixmust have the general form

T (ε) = t(ε) + τ(ε)σ · S (11.37)

where the electronic spin-flip transitions are acccounted for by the scalar productterm.

There are two kinds of processes contributing to the elastic scattering off themagnetic impurity: a) spin conserving transitions and b) spin-flip transitions.For the spin-conserving case, we have:

W (k, ↑M → k′, ↑M) =2π

h

∑

M

wM |T (k′, ↑M | k, ↑M) | 2δ(εk − ε′k) (11.38)

For the spin-flip case in the first Born approximation to T we write:

W (k ↑M → k′ ↓ ,M + 1) =2π

h

∑

M

wM |T (k′ ↑M | k ↓ M + 1)| 2δ(εk − ε′k)

(11.39)


Exercise 11.4Show that from (11.37) and the matrix elements of S (see Eqs. (4.6)) one ob-tains:

T↑↑ = t+ τM

T↑↓ = τ√

S(S + 1) −M(M + 1) (11.40)

The first two terms in Eq. (11.35) correspond respectively to the first and thesecond Born approximation. To calculate the collision time τc in the first Bornapproximation we add the transition probabilities for both independent channels(spin-conserving and spin-flip), average over all initial impurity spin orientations(assumed equally probable), and sum over all final states on the energy shell,under the restriction that total spin be conserved:

1

τc=

2π

hρ(εk)

∑

M

wM

(

|T (k′ ↑ M |k ↑M)|2εk=εk′

+ |T (k′ ↓ M + 1|k ↑M)|2εk=εk′

)

(11.41)

Using Eq. (11.4.2) we get:

1

τc=

2π

hρ(εk)

[

|t(εk)|2 + |τ(εk)|2S(S + 1)]

(11.42)

Exercise 11.5Prove Eq. (11.42)

Since in the first Born approximation T = V = Hsd, we see that

τ (1) =−J2N

, t(1) = 0. (11.43)

where we take J = constant. We can now substitute for τc in Eq. (11.31), andinvert σ to find the resistivity:

Exercise 11.6Show that if the T matrix is calculated to first order in Hsd we get for theresistivity

ρ(1) =3

2

mπ

e2h

V

EF

J2

4N2S(S + 1) cimp (11.44)

We must include in ρ(1) the factor Nimp=total number of impurities, so thatthe resistivity is proportional to their atomic concentration cimp = Nimp/N .We turn now to the second order contribution. The initial state is in both casesan incoming electron in state k ↑ and the impurity in state (S,M). We rec-ognize now that there are two possible intermediate states, which generate twopossible transitions to the final state of the electron:


(a) in the first case an incoming electron in state (k ↑) interacts with the impu-rity in state (S,M) and is annihilated, while an intermediate electron in state(k′′ ↑ (↓) is created and the impurity either stays in the same state or makes atransition to the state (S,M + 1) to conserve total spin. The intermediate elec-tron is further annihilated by the exchange interaction, the impurity recoversits initial state, and the outgoing electron is created in state (k′, ↑).

(b) in the second case, the incoming electron does not interact first with theimpurity but continues propagating, while the interaction creates an electron-hole pair, with the electron occuppying the final state (k′ ↑), while the hole canbe in either state (k′′, ↑ , ↓). For the spin-conserving case, we first annihilate anelectron in state (k′′ ↑) and create another in state (k′ ↑), while the impuritystate does not change. The interaction acts again to ahhihilate the incomingstate (k′ ↑) and create an intermediate electron state (k′′ ↑). In the spin-flip casethe interaction first annihilates an electron in state (k′′ ↓) and creates anotherone in the final state (k′ ↑), while the second interaction event annhilates theinitial electron state and creates the intermediate state (k′′ ↓), so the total spinchange of the electron band is +1, which must be compensated by a transitionof the impurity from Sz = M to Sz = M − 1. The matrix elements of Hsd

for the spin-flip transition are of course those proportional to the transversecomponents S(+,−).

We calculate now the second order contribution to T . Thanks to the decom-position (11.37) it is enough to calculate T↑↑. For the two processes describedabove we find respectively:

T (2)a (k′ ↑ , k ↑) =

J2

4N2〈M |(Sz)2|M 〉

〈 0 ak′↑|a†k′↑ak′′↑a†k′′↑ak↑|a†k↑|0〉

εk − εk′′ + iη

+J2

4N2〈M |S−S+ |M 〉

〈 0 ak′↑|a†k′↑ak′′↓a†k′′↓ak↑|a†k↑|0〉

εk − εk′′ + iη(11.45)

T(2)b (k′ ↑ , k ↑) =

J2

4N2〈M |(Sz)2|M 〉

〈 0 ak′↑|ak′′↑ak↑a†k′′↑ak↑|a†k↑|0〉

εk′′ − εk + iη

+J2

4N2〈M |S+S− |M 〉

〈 0 ak′↑|a†k′′↓a†k′↑ak↑a

†k′↑|ak′′↓|ak↑|0〉

εk′′ − εk + iη(11.46)

We substitute above the matrix elements of S(+,−) from Eq. (4.6) and the num-ber operator of the intermediate state by its statistical average at temperatureT , so that the matrix elements of T for both types of processes are:

Ta(εk) =

(

J

2N

)2

[S(S + 1) −M ]

∫ ∞

−∞

ρ(E)1 − f(E)

ε−E − iηdE


Tb(εk) =

(

J

2N

)2

[S(S + 1) +M ]

∫ ∞

−∞

ρ(E)f(E)

ε−E − iηdE (11.47)

Exercise 11.7Obtain Eqs. (11.46) and (11.47).

The terms proportional to M are those originating from the τ term in T , so thatwe get the real part of the second order contribution to the matrix elements ofτ as

Re τ(ε+ iη)(2) = −(

J

2N

)2 ∫ ∞

−∞

ρ(E)P

(

1 − 2f(E)

ε−E − iη

)

dE (11.48)

where P (.) is the principal value operator. We concentrate on the real part ofτ (2) because that will generate the singularity we are looking for. To simplify,we take a constant density of states distribution:

ρ(E) =

ρ , −D < E < D0 , |E| > D

(11.49)

Notice that energies are now measured from the Fermi level.Call P the principal part integral:

P (ε) =

∫ D

−D

1 − 2f(E)

ε−EdE (11.50)

We integrate by parts:

P (ε) = − [ ln | ε−E | (1 − 2f(E)) ]D−D − 2

∫ D

−D

ln |ε−E| f ′(E) dE (11.51)

We take D kBT , so that f(−D) ≈ 1 and f(D) ≈ 0. The first term of (11.51)is then

−2 ln | ε−E | (11.52)

For the second term we must consider separately two different cases, accordingto whether |ε| > or < kBT .

In the first case, since f ′(E) ≈ 0 if |E| > kBT we neglect E with respect toε in the argument of the logarithm, so the second term is:

−2 ln |ε|∫ D

−D

f ′(E) dE ≈ 2 ln | ε | (11.53)

If we add this to the first term we have that, for | ε | > kBT ,

P (ε) = 2 lnE

D(11.54)


Let us look, for the second case, that is |ε| < kBT , at the second term in(11.51). In the integrand we can neglect ε with respect to E in the argument ofthe logarithm, so that the integral is:

−2

∫ D

−D

ln |ε−E| f ′(E) dE ≈ −2

∫ ∞

−∞

(

ln|E|kBT

+ ln kBT

)

f ′(E) dE

≈ 2 ln kBT − 2 ln (2eγ/π) (11.55)


We add the first and second order contributions to Re (τ) we have just found:

τ (1) +Re(

τ (2))

≈ − J

2N

(

1 +Jρ

Nln

∆

D

)

(11.56)

where

∆ =

|ε| if |ε| > kBTkBT if |ε| < kBT

(11.57)

Then we get the collision time, including the first order term and the singularpart of the second order term just calculated, with |ε| ≤ kB T :

1

τc(ε)=

2πρ

hNiS(S + 1)

(

J

2N

)2(

1 +Jρ

Nln kB T/D

)2

(11.58)

where we included the factor Ni = the number of impurities to get the totaltransition probability. For the resistivity, then, we find:

ρ(2)(T ) =3m

2e2V

4NJ2cimp

π

EF hS(S + 1)

(

1 + 2Jρ

Nln ∆/D

)

(11.59)

We must add this contribution to the phonon one and to the residual resistivityarising from elastic scattering processes due to fixed defects and non-magneticimpurities. The contribution to the resistivity from the electron-electron colli-sions, which is of the order of (kBT/EF )2 [10] is negligible at low T , where theKondo resistance is important. Then we can write the resistivity at low T as

R(T ) = const.+A

(

kBT

ΘD

)5

+B lnkBT

D(11.60)

Exercise 11.9Show that in the case J < 0 there is a minimum of the resistivity at a finite T .


11.4.3 Higher order contributions

Abrikosov [11] made a diagrammatic expansion of the T matrix to all orderswith the use of a drone fermion representation of the spin operators due toEliashberg [12]. The sum of the most divergent terms for the spin-dependentpart τ yields for the matrix element

τ(ε) =−J2N

1

1 − JρN ln ∆

D

(11.61)

and we can take ∆ ≈ kB T as we mentioned before. The non-flip part has aweaker divergence and we can leave it out.

For J > 0 (11.61) makes a negligible contribution to the resistance as T → 0.For J < 0, which is the case in Cu −Mn, there is a divergence at the Kondotemperature, which in this formulation is defined as

kBTK = De−N/|J|ρ (11.62)

At this temperature the denominator in (11.61) vanishes. (It is important tonotice that TK does not depend on the impurity concentration.)

Since the transition probability diverges, the collision time τc → 0 and ac-cordingly σ → 0 and ρ→ ∞ as T TK .

Although there is no experimentally observed divergence of R(TK), subse-quent work supports the existence of a characteristic temperature of the orderof the above one. An exact theoretical solution of the Kondo problem was ob-tained by Andrei et.al. [13] and by Tsvelick and Wiegmann[14].

The picture which has emerged from a large body of experimental and the-oretical work is that for J < 0 and T TK the impurity does not sustain aspin, because under the influence of the AF exchange interaction, the resonantbehaviour of the T matrix at low temperature localizes an electron cloud ofopposite spin around the impurity which compensates the spin of the latter. Asthe temperature grows above TK this magnetic screening disappears, and thesusceptibility recovers the Curie behaviour characteristic of a local spin.[7]

References


2. Yosida, Kei (1998) “Theory of Magnetism”, Springer.

3. Peierls, R. E. , (1955) “Quantum Theory of Solids”, chap. 6. Re-editedin 2001 by Clarendon Press, Cambridge.

4. de Haas, W. J. et al. (1934) Physica 1, 1115.

5. Sarachik et al. (1964) Phys. Rev. A 135, 1041.

6. Kondo, J. (1964) Prog. Theor. Phys. 32, 37.


7. Hewson, A. C. (1997) “The Kondo problem to heavy fermions”, Cam-bridge University Press.

8. Peierls, R. E., loc. cit., Chap. 6.

9. Peierls, R. E., loc. cit., equation (4.42).

10. Peierls, R. E., loc. cit., Chap. 6.

11. Abrikosov, A. A. (1985) Physics 2, 5.

12. Eliashberg, G. M. (1962) JETP 42, 1658.

13. Andrei, N. et al. (1983) Rev. Mod. Phys. 55, 331.

14. Tsvelick, A. M. and Wiegmann, P. B. (1983) Adv. Phys. 32, 953.

Chapter 12

Low Dimensions

12.1 Introduction

We shall start the discussion of the theory of low dimensional systems with asurvey of the theorem of Mermin and Wagner [1] on the absence of long rangeorder (LRO) in magnetic systems. A similar theorem was proved by Hohenberg[2] on superfluids and superconductors. The contents of this theorem as appliedto systems of localized spins is the following:

Theorem 12.1 Mermin Wagner theorem.An infinite d dimensional lattice of localized spins cannot have LRO at anyfinite temperature for d < 3 if the effective exchange interactions among spinsare isotropic in spin space and of finite range.

Since the conditions above are precisely satisfied by the Heisenberg model thatwe have used for most of the theoretical description of magnetic insulators untilnow, it turns out that we cannot extend those results to lower dimensions. Wehave already arrived at this conclusion when we found divergences upon theapplication of the free-spin-wave approximation (FSWA) and the RPA to lowdimensional systems. What we present here is a general result which does notdepend on any approximation.

Let us make a few remarks at this point:

(i) We cannot discard LRO at zero T in the isotropic Heisenberg model. Itturns out that low dimensional ferromagnets exhibit LRO at T = 0, but anti-ferromagnets do not;

(ii) When infinite range interactions are considered, e.g. the dipolar interac-tions or the RKKY interactions, the theorem does not apply and we might findfinite critical temperatures even in low dimensions;

(iii) When we incorporate anisotropy into the model (non-isotropic exchange

277

278 CHAPTER 12. LOW DIMENSIONS

constants, dipole-dipole interactions, single-ion anisotropy, etc.) we again ex-pect to find cases when low dimensional systems will exhibit LRO.

Although rigourously speaking there are no realizations of 1 or 2 dimen-sional systems, in many practical cases one can be very close to this situation,and the theoretical description of the ideal system is extremely useful to an-alyze the experimental results in cases of quasi-two or quasi-one-dimensionalsystems. In practice, these are lattices in which some family of lines or planescontain strongly interacting spins which instead couple weakly with spins onother lines or planes. A paradigm of quasi-two dimensional systems is providedby the highly anisotropic compounds which are parents of the high-Tc supercon-ducting perovskites. In particular, La2CuO4 and Y Ba2Cu3O6 show fairly largeNeel temperatures TN (≈ 300−400K) and strongly anisotropic AF correlations.The latter indicate that the effective exchange coupling within Cu++ ions in theCuO2 planes is unusually large (' 0.1 eV ) while the exchange anisotropy ratioε = J⊥/J‖ 1 (see ref. [3] and references therein). The parameter ε controlsthe cross-over from 3d (ε ' 1) to 2d as ε→ 0 or indeed to 1d as ε→ ∞.K2NiF4 is another quasi- 2d AFM with perovskite structure [4]. Experimentsshow long range AFM correlations on the NiF2 planes at temperatures wellabove the 3d Neel temperature.

In 1d we mention the compound (C6H11NH3)CuBr (CHAB) which behaveslike an ideal 1d easy-plane FM with S = 1/2 [5] and the 1d antiferromagnet[(CH3)4N ] [MnCl3] (TMMC), which has chains of M2+

n ions (S = 5/2) [6].

12.2 Proof of Mermin Wagner theorem

Following the original paper [1] we start by proving Bogoliubov inequality [7].

12.2.1 Bogoliubov inequality

Let | i〉 be a complete orthonormalized set of eigenfunctions of a given Hamil-tonian H , and let A,B be two operators, of which we only assume that all theirmatrix elements in the basis above are well defined and bounded. Then wedefine an inner product of that pair of operators as:

Definition:

(A,B) ≡∑

i6=j

〈i|A | j〉∗〈i | A | j〉Wi −Wj

Ej −Ei, (12.1)

where En is an eigenvalue of H , Wn = e−βEn/ρ and

ρ = Tr(

e−βH)

(12.2)

with β = 1/kBT . Notice that we exclude the terms with vanishing denominatorfrom the sum (12.1).

12.2. PROOF OF MERMIN WAGNER THEOREM 279

Exercise 12.1Prove that this inner product has the following properties:

(u, v) i = (v, u)∗

;

(u, a1v1 + a2v2) = a1 (u, v1) + a2 (u, v2) ;

(u, u) ≥ 0 .

As a consequence it satisfies the Cauchy-Schwartz inequality:

(A,A)(B,B) ≥| (A,B) |2 (12.3)

The following inequality holds:

0 ≤ Wi −Wj

Ei −Ej≤ (Wi +Wj)β/2 (12.4)

and therefore

0 ≤ (A,A) ≤ β

2〈[

A,A†]

+〉 (12.5)

where [ ·, · ]+ = anticommutator. Now we choose B =[

C†, H]

and then

(A,B) = 〈[

C†, A†]

〉(B,B) = 〈

[

C†, [ H,C ]]

〉 (12.6)

From (12.1) to (12.6) we can prove that:

1

2〈[

A,A†]

+〉〈[

[ C,H ] , C†]

〉 ≥ kBT | 〈[ C,A ]〉 |2 (12.7)

(Bogoliubov inequality) [7].


12.2.2 Application to the Heisenberg model

Let us consider again the Hamiltonian

H = −∑

R,R′

J(R −R′)S(R) · S(R′) − h∑

R

Sz(R)e−iQ·R (12.8)

In the FM case we take Q = 0 ; in an AFM the phase φ ≡ Q ·R = (2n + 1)πwhen R connects points on different sublattices, and φ = 2nπ in the oppositecase. The translational symmetry, for an infinite crystalline sample, is alreadyimplicit in the form of J . We assume J is an even function of its argument.The Fourier transformed spin operators

Sα(q) =∑

R

e−iq·RSα(R) (12.9)


where α = +,− or z, satisfy the commutation relations

[

S+(k),S−(k′)]

= 2Sz(k + k′)[

S±(k),Sz(k′)]

= ∓S±(k + k′) . (12.10)

We make now the following choices for the operators A and C in Eq. (12.7):

A = S−(−k −Q) , C = S+(k) (12.11)

The statistical average of the double commutator in (12.7) is:

Dk(Q) = 〈[

[ C,H ] , C†]

〉

=1

N

∑

k′

(Jk′ − Jk′+k)〈S−k′S

+−k′ + S+

k′S−−k′ + 4Sz

k′Sz−k′〉 + 2γhsz(−Q)

(12.12)

where

sz(Q) =1

N

∑

R

〈Sz(R)〉eiQ·R (12.13)

and we have explicitly used the property J(−q) = J(q).We remark that in the FM case (Q = 0), sz(0) ∝ average uniform magne-

tization, while in the AFM case

|sz(Q)| = (1/2)|〈Saz − Sb

z〉| (12.14)

( a and b are the up- and down-sublattices) which is > 0 in the Neel phase.As to the applied field, it is a uniform field in the FM case and a staggered

field in the AFM one.With our choice of operators Bogoliubov inequality reads:

1

2〈[

S+(k + Q) , S−(−k −Q)]

+〉 ≥ 4kBTN

2sz(Q)2

Dk(Q)(12.15)

According to Eq. (9.6) Dk(Q) is a norm squared, so that | Dk(Q) |= Dk(Q) andwe can use the inequality

|∑

n

an | ≤∑

n

| an | . (12.16)

to find an upper bound for D.We define the quantity

∆ = S(S + 1)∑

R

R2 | J(R) | (12.17)


Dk(Q) ≤ Nk2∆ +N | hsz(Q) | (12.18)

12.2. PROOF OF MERMIN WAGNER THEOREM 281

We now substitute D in (12.15) by the upper bound above, and then sum bothsides of the resultant inequality over k. Upon use of the identity

∑

k′

S(k′) · S(−k′) = N2S(S + 1) (12.19)

we verify easily that (12.19) is an upper bound for the sum of the left hand sideof Eq. (12.15). Then we obtain:

sz(Q)2 ≤ 2βS(S + 1)

ΦQ(12.20)

where

ΦQ =1

N

∑

k

1

∆k2+ | hsz(Q) | (12.21)

For V → ∞ and a lattice of ν dimensions Eq. (12.21) reads:

ΦQ =1

ρ(2π)ν

∫

dνk

1

∆ + | hsz(Q) |

(12.22)

where ρ = N/V and we integrate inside the first BZ.Since sz(Q)2 is inversely proportional to Φ we reinforce inequality (12.20) by

integrating in Eq. (12.22) over a smaller domain, since the integrand is positivedefinite. This does not alter any conclusion we may reach on the convergenceof the integral leading to Φ, since it is clear that the critical region in the limith→ 0 is the neighbourhood of the origin in k space. Let us then integrate overa spherical region with center at the origin and radius k0 equal to the minimumdistance from the origin to the boundary of the first BZ.

Two dimensions:

s2z ≤ 2πS(S + 1)ρ

kBT

∆

ln(1 + ∆/ | hsz |) (12.23)

which in the limit h→ 0 yields

| sz |≤ Const

T 1/2

1

| ln | h ||1/2. (12.24)

One dimension:

s2z ≤ 4S(S + 1)πρ∆1/2 | hsz |1/2

kBT arctan(

∆1/2k0/ | hsz |1/2) (12.25)

and in the limit h→ 0 we get:

sz ≤ Const

T 2/3| h |1/3 (12.26)


Exercise 12.4Verify Eqs. (12.22) to (12.26).

Equations (12.24) and (12.26) are the main results of Mermin and Wagner.They imply that in the h→ 0 limit the LRO parameter sz → 0 for ν < 3 underthe assumptions of short range, isotropic exchange interactions.

As regards the first assumption (short range), if the parameter ∆ definedin (12.17) diverges we arrive at no conclusion. If J(R) ≈ R−α for large R thecondition for convergence of the integral in Eq. (12.17) is α > ν + 2. Conse-quently we cannot exclude the existence of LRO in low dimensions for dipolaror RKKY interactions. The divergence of Φ, according to the definition (12.21)is due to the behaviour of the denominator for small k. The above conditionis the present criterion for what we shall consider long range effective exchangeinteractions. The same condition was found by Dyson [8] for the existence of aphase transition in the one dimensional Ising model, as we shall discuss furtheron in this chapter.

As to the second condition (isotropy), we remark that the denominator is thesmall k expansion of the dispersion relation for an isotropic FM in the spin waveapproximation. We know that if an anisotropy of the right kind is present thedispersion relation acquires a gap for k = 0, and this automatically invalidatesMW theorem, since Φ is finite in the h→ 0 limit.

12.3 Dipolar interactions in low dimensions

We address now the exchange-dipolar low d magnetic insulator as a particularlyinteresting system because dipolar forces exhibit both long range and anisotropyand they are always present when there are magnetic moments, while otherkinds of anisotropic forces depend on special characteristics of the system, andin many cases are absent.

Let us consider a ferromagnetic chain. If we choose the z axis along thechain, then xn = yn = 0 , zn = na.

The dipolar tensor

Dαβ(R) ≡ 1

R3

(

δαβ − 3RαRβ

R2

)

(12.27)

is diagonal for the chain, D, and

Dxx(n,m) = Dyy(n,m) =1

a3 | n−m |3

Dzz(n,m) = − 2

a3 | n−m |3 (12.28)

The classical system has energy

Uclass =γ2

2a3

∑

n,m

′ Dαβ(n,m)SαnS

βm (12.29)

12.3. DIPOLAR INTERACTIONS IN LOW DIMENSIONS 283

Consider first a state with all spins aligned along the chain. Performing thesums we get, for N spins

U‖class

N=γ2ζ(3)

a3[S(S + 1) − 3S2

z ] (12.30)

where

ζ(p) =

∞∑

1

1

np(12.31)

is Rieman n’s ζ function of order p.For spins perpendicular to the chain we choose z as axis of quantization and

we take the chain along the x axis. Therefore, xn = na, yn = zn = 0 and

U⊥class

N=γ2ζ(3)

a3[S(S + 1) − 3S2

x] (12.32)

We calculate only the dipolar part of the energy since the exchange part is thesame in both cases. Eqs. (12.29) and (12.32) are special cases of

Udipclass

N=γ2ζ(3)S(S + 1)

a3

[

1 − 3S

S + 1cos2 φ

]

(12.33)

valid for any uniform orientation of the spins, and where φ is the angle themagnetization makes with the chain. φ = 0 minimizes (12.33), so that theWeiss state is aligned along the chain.

Consider now the quantum case. From Chap. 6 we recall the commutator

[S+a , Hdip] = −γ

2

2S+

a

∑

m

′SzmDzz

am

− γ2

4Sz

a

∑

m

′S+mDzz

am − 3γ2

4Sz

a

∑

m

′S−mB

∗am

+3γ2

2S+

a

∑

m

′S+mFam − 3γ2

2Sz

a

∑

m

′SzmF

∗am

+3γ2

4S+

a

∑

m

′S−mF

∗am (12.34)

The coefficients B and F above vanish if spins are polarized along the chain.Accordingly, in this case there are no zero point fluctuations of the transversecomponents of the total spin and the z component of the total spin is conservedat T = 0.

For spin sites a, b, the double-time retarded Zubarev Green’s function satis-fies the RPA equation

ωG+−ab (ω) =

〈Sza〉δab

π− 2

∑

m6=a

Jma

〈Sza〉G+−

mb (ω)


− 〈Szm〉G+−

ab (ω)

− γ2G+−ab (ω)

∑

m6=a

Dzzma〈Sz

m〉

− γ2

2〈Sz

a〉∑

m6=a

DzzmaG

+−mb (ω) + γBG+−

ab (ω) (12.35)

We see that for spins polarized along the chain there is no coupling of G+− withG−−.

Fourier transforming Eq. (12.35) to k space we obtain

G+−k (ω) =

σ/π

ω −Ek(12.36)

where the dispersion relation is

Ek = 2σ(

J(0) − J(k))

+ γB − γ2σ

(

Dzz(0) +Dzz(k)

2

)

(12.37)

with the same notation as in Chap. 6. In order to obtain an explicit expressionfor Ek we need the Fourier transform

Dzz(k) = − 2

a3

∑

n

′ eikna

| n |3 ≡ − 2

a3A(q) (12.38)

with q = ka.Let us describe now Ewald’s method [9, 10] for the calculation of lattice

sums like A(q).As a first step, we make use of a representation of | n |−3 based on the

definition of the Γ function:

x−α =1

Γ(α/2)

∫ ∞

0

t(α/2−1)e−tx2

dt (12.39)

It turns out convenient to include a small shift in the coordinate n : n→ n+r,and to take the limit r → 0 at the end of the calculation. Then

A(q) = limr→0

1

Γ(3/2)

∫ ∞

0

dt∑

n6=0

eiq(n+r)e−t(n+r)2t1/2 (12.40)

We now add and subtract the n = 0 term in the sum above:

A(q) = limr→0

[

2√π

∫ ∞

0

dt∑

n

eiq(n+r)e−t(n+r)2t1/2 − eiqr

r3

]

(12.41)

The integrand above is a periodic function and it can be expanded in a Fourierseries in reciprocal 1d space. Let us write

F (r) =

∞∑

n=−∞

eiq(n+r)e−t(n+r)2 =∑

p

e−2iπprFp (12.42)

12.3. DIPOLAR INTERACTIONS IN LOW DIMENSIONS 285

where

Fp =

∫ 1

0

dx F (x)e2iπpx =

∫ ∞

−∞

dx eiGxe−tx2

=√π t−1/2e−G2/4t (12.43)

where G = 2πp+ q. We finally arrive at

∑

n

eiq(n+r)e−t(n+r)2 =∑

p

e−2πpr√π t−1/2e−(2πp+q)2/4t (12.44)

This identity is known as the Jacobi imaginary transformation of the ζ func-tion [10]. It can be readily extended to multiple summations, i.e. higher dimen-sions. In order to get a final expression for A(q) (Eq. (12.41)) we must performthe integration over t and the limit r → 0. Both sides of Eq. (12.44) are differentrepresentations of the same function of t. For small t the convergence of theintegral (12.41) near t = 0 is better if we use the representation of the r. h.s. of Eq. (12.44), while the opposite is true for large values of t. We introducetherefore Ewald’s separation parameter ξ, and we write

A(q) = limr→0

2

∞∑

p=−∞

∫ ξ

0

e−(2πp+q)2/4te−2πprdt +

2√π

∞∑

n=−∞

∫ ∞

ξ

eiq(n+r)e−t(n+r)2t1/2dt− eiqr

r3

(12.45)

The n = 0 term in Eq. (12.45) is

eiqr

∫ ∞

0

−∫ ξ

0

e−tr2

t1/2dt (12.46)

The first integral cancels exactly the subtraction term, and we are left with thefinite correction term

−4

3

ξ3√π

The integral in the reciprocal lattice sum of Eq. (12.45) (first term) ccan betransformed by the change of variables t = 1/z into the exponential integralof order 2, E2 [11] of the corresponding argument. The integral in the directlattice sum (second term) is transformed by the change t = z2 into expressionsinvolving Gaussians and the error function:

A(q) = Alatt(q) +Arec(q) , where

Alatt(q) =2√π

∑

n6=0

eiqn

(

ξ1/2

n2e−ξn2

+

√π

2n3erfc(ξ1/2 | n | )

)

− 4

3

ξ3√π

Arec(q) = 2ξ

∞∑

p=−∞

E2

(

(2πp+ q)2

4ξ

)

(12.47)


The p = 0 term in Arec(q) has a logarithmic sigularity when q → 0. In order todisplay the form of this singularity we use the series expansion of E2 for smallargument [11] and find:

2ξE2(q2/4ξ) = 2ξ

(q2/4ξ)( log (q2/4ξ) − ψ(2) ) −∞∑

m=0,6=1

(q2/4ξ)m

m!(m− 1)

(12.48)

where ψ(x) is the digamma function [11]. One verifies that as ξ → 0 thereciprocal space sum vanishes and the direct lattice sum goes back to the originalexpression for A(k). Ewald’s method yields two fast convergent series. The sumof the two series in (12.47) is identical to the original expression (12.45) andaccordingly it is independent on the value of ξ , which can be chosen to optimizethe series convergence or simplify the final expressions. In particular, note thatthe singular term in Eq. (12.48) is independent on ξ. At q = 0 we have fromEq. 12.37:

E0 = γB + 6γ2σζ(3)/a3 , (12.49)

so that even in the absence of an external field (B = 0) there is a finite gapin the spectrum which warrants LRO. The presence of this gap implies thatGoldstone’s theorem does not apply in this case. This is in agreement with thefact that the symmetry group of the ground state, for spins along the chain,is discrete. In effect we have two degenerate ground states, with spins alignedin one of the two directions ±z. The group consists of the identity and theinversion. The absence of a continuous symmetry group implies the possibilityof a finite excitation energy from the ground state to the first excited one.

12.3.1 Dipole-exchange cross-over

From the dispersion relation (12.37) we conclude that at small q (i.e. at largedistances) the dipolar terms dominate, while the opposite is true when q isa sizeable fraction of π/a. The relevant parameter is the ratio of the typicaldipolar and exchange energies which we define as the dipolar anisotropy Ed:

Ed =γ2

Ja3

We can rewrite Eq for q small as

Eq

SJ= q2 +Ed

(

6ζ(3) −Aq2 log q2 +Bq2)

+ O(q3) (12.50)

where A,B = numerical coefficients of order 1. The first term in q2 on the r.h. s. of Eq. 12.50 is the exchange contribution, while terms proportional to Ed

are either dipolar-exchange or pure dipolar ones. The dipolar terms dominatedue to the logarithmic factor in the limit q → 0. When Ed | log q2c |≈ 1 the

12.4. ONE DIMENSIONAL INSTABILITIES 287

dipolar and exchange terms become comparable, and there is a cross-over to theexchange dominated regime. Typical values of Ed are 10−2 to 10−4. The cross-over qc is exponentially small, so for most values of q inside the BZ exchangedominates the dynamics of the one-dimensional magnons. However, the groundstate properties, in particular the existence of a gap and the polarization ofspins, are determined by the dipolar interactions, since they are controlled bythe q → 0 limit.

12.4 One dimensional instabilities

In spite of these results one must be cautious before asserting the existence ofLRO in the FM chain. There is a well known argument due to R. E. Peierls[12] in this respect. The basic idea is that one possible source of disorder atlow (but not zero) T is the spontaneous breaking of the chain into FM domainsseparated by kinks. For spins polarized basically along the chain at very lowT , all spins before a kink located at the site ni would point in, say, the +zdirection, while for n > ni they would point in the −z direction. These defectswill be randomly distributed along the chain on a set ni of points. Each suchdefect increases the interaction energy, but their contribution to the entropydecreases the Helmholtz free energy

F = U − TS (12.51)

and we must determine the minimum of F in the presence of a distribution ofkinks. The crucial hypothesis in Peierls’ argument is the assumption of shortrange of J(n). In such a case let us consider the change in the internal energy∆U upon introducing N0 N kinks. Since they are sparsely distributed alongthe chain, they are separated by an average distance much greater than therange of J(n) and therefore they do not interfere with one another. Then ∆Uis proportional to N0:

∆U

N= C

N0

N≡ Cρ , (12.52)

ρ = N0/N being the average concentration of kinks and C > 0. If the probabilityof a kink being localized at any given point, very far from other kinks, is someconstant P , then the total probability of findingN0 kinks is proportional toWP ,where W = number of ways of selecting N0 out of N sites is the combinatorialnumber

W =

(

N0

N

)

(12.53)

We approximate W , for (N0, N) → ∞, with Stirling’s formula, obtaining forthe entropy the estimate

S = kB logW ≈ kBN0 log ρ (12.54)


and for ∆F = F (T ) − F (0)

∆F

N=

∆U

N− TS = Cρ− kBTρ log ρ (12.55)

neglecting terms O(1/N). We now minimize ∆F with respect to ρ and find

ρ(T ) = e−C/kBT (12.56)

So under the present assumptions, there will be a finite concentration of kinks,and consequently no LRO, at any finite T in the FM chain. The argumentapplies both to the Ising and the Heisenberg model. A similar argument ledPeierls [12] to prove that in 2d the Ising FM does have a phase transition at afinite Tc.

However, no long range forces have been taken into account in the aboveargument. The problem of the existence of a phase transition in the presenceof long range forces has been considered within the FM Ising model, defined bythe Hamiltonian

H = −∑

i>j

J(i− j)µiµj , µi = ±1 , ∀i (12.57)

withJ(n) > 0 , ∀n . (12.58)

For this case it can be proven [13] that the infinite system is a well defined limitof the finite one, with consistent definitions of the thermodynamic averages,provided the zero order moment of the coupling satisfies

M0 ≡∞∑

n=1

J(n) <∞ (12.59)

Exercise 12.5Prove that when M0 = ∞ in the Ising model there is an infinite energy gap sep-arating the Weiss ground state from the excited states. Therefore at any finiteT the system will remain ordered: there is no phase transition to a disorderedstate.

If M0 satisfies (12.59) the system will be disordered at any finite T , as wehave just seen [14]. A theorem by Ruelle [15] proves that an Ising chain whichsatisfies Eqs. (12.57) and (12.58) will be disordered at any finite T provided

M1 ≡∞∑

n=1

nJ(n) <∞ (12.60)

Exercise 12.6Prove that 2M1 is the energy for creating one kink in an Ising FM chain.

Kac and Thompson [16] made the

12.5. ANTIFERROMAGNETIC CHAIN 289

Conjecture 1 A 1d system obeying (12.57) with

M0 <∞, M1 = ∞

has a phase transition at some finite T .

From this follows

Corollary 1 A 1d system obeying (12.57) with

J(n) = n−α (12.61)

has a phase transition at a finite T if and only if

1 < α ≤ 2 (12.62)

Based on the preceding results Dyson [8] proved that an Ising system satisfyingEq. (12.61) has a phase transition if 1 < α < 2. Therefore there is no phasetransition at finite T in an Ising system with dipolar interactions.

Dyson conjectures that the same is true for the Heisenberg case.In conclusion, we probably should discard the existence of LRO in a FM

chain at finite T . However, as we indicated at the beginning of this chapter,one finds excitations similar to one dimensional magnons in many experimentson quasi-one dimensional ferromagnets, indicating the existence of FM orderover distances larger or at least comparable with the mean free path of theseexcitations, even at fairly high T .

12.5 Antiferromagnetic chain

We consider now a chain of spins with AFM short range exchange togetherwith dipole-dipole magnetic interactions. We know that the one-dimensionalHeisenberg AFM does not have LRO even at zero temperature. Just as for aFM, we shall ask ourselves whether dipolar interactions could lead to LRO [17].We write again

H = −∑

a6=b

JabSa · Sb +∑

a6=b

Sαa S

βb Dαβ

ab (12.63)

with Jab < 0. At low T we take the Neel state as a good approximation tothe ground state when exchange prevails. If dipolar interactions dominate, theminimum energy configuration is the FM ordering along the chain. To calculatethe classical energy consider all spins substituted by c-vectors. We write thespin at point zn = na along the chain as:

Sn = S0eik0na , S2

0 = S(S + 1) (12.64)

where q0 ≡ k0a satisfies the condition

q0 = (2m+ 1)π , m = integer (12.65)


in the Neel state. The total classical energy with the assumption (12.64) is

Uclass(q0)

NS(S + 1)=γ2

2

∑

n6=0

eiq0nDαα(n) −∑

n6=0

eiq0nJ(n) (12.66)

where α denotes the spin polarization axis. We consider now the longitudinaland transverse polarizations separately.

S0 ‖ chain.

For this case, in which the quantization axis z is along the chain,

U‖class(q0)

NS(S + 1)= γ22Dzz(q0) − Jq0 (12.67)

With the condition (12.65) on q0 we find:

Dzz(q0) =3ζ(3)

a3(12.68)

and

U‖class(q0)

NS(S + 1)=

3ζ(3)

a3− Jq0 (12.69)

For q0 = 0 (FM order) we have

Dzz(0) = −4ζ(3)

a3(12.70)

and

U‖class(0)

NS(S + 1)= −2γ2ζ(3)

a3− J0 (12.71)

which is the dipolar-dominated configuration as we already know (see Eq. (12.33)).

S0 ⊥ chain.

Now we take as in the FM case the x axis along the chain, and z along thequantization axis. Then

Dzz(q0) = −3ζ(3)

2a3(12.72)

andU⊥

class(q0)

NS(S + 1)= −3γ2ζ(3)

4a3− Jq0 (12.73)

We can compare the dipolar dominated configuration of Eq. (12.71) with theexchange dominated one from Eq. (12.73). If we limit ourselves to only first

12.6. RPA FOR THE AFM CHAIN 291

n.n. exchange interactions, the critical dipolar anisotropy Ecd for the cross-over

from exchange- to dipolar-dominated regime is

Ecd ≡ γ2

Ja3=

16

5ζ(3)≈ 2.66 (12.74)

For Ed > (<)Ecd the FM (AFM) configuration is more stable.

12.6 RPA for the AFM chain

12.6.1 Exchange dominated regime

We assume that the ground state has the AFM configuration

〈Szn〉 = σeiq0n (12.75)

with z perpendicular to the chain, which we take along the x axis. Followingthe same procedure as in Chap. 6, we obtain the algebraic equation satisfied bythe retarded Green’s function G+−(q, ω) in the RPA [17]:

ω

σG+−(q, ω) =

1

π− 2[ Jq+q0 − Jq0 ]G+−(q, ω)

−γ2[ Dzz(q0) +1

2Dzz(q + q0) ]G+−(q + q0, ω) −

−3γ2

2B∗(q + q0)G

−−(q + q0, ω) (12.76)

where

B(q) =∑

n

′ (xn − iyn)2

(x2n + y2

n + z2n)5/2

eiqn (12.77)

and we have xn = na, yn = zn = 0 , ∀n. Then B(q) = B∗(q) = Dzz(q). Let uschoose the spin at b as ↑. A look at Eq. (12.63) tells us that a spin reversal at agiven site couples with spin reversals in the other sublattice through exchangeand also to spin ascending transitions due to dipolar forces, so that now wemust deal with four Green’s functions.

Let us choose the origin of x coordinates along the chain at an ↑ site. Thealternative choice of origin at the ↓ sublattice only changes the signs of all theFourier transforms, which does not alter the dispersion relations. We write nowthe equations for G−−(q, ω) :

ω

σG−−(q, ω) = 2[ Jq+q0 − Jq0 ]G−−(q, ω) +

γ2[ Dzz(q0) +1

2Dzz(q + q0) ]G−−(q + q0, ω) +

3γ2

2B(q + q0)G

+−(q + q0, ω) (12.78)


To simplify the notation we define

ck = 2(Jk+q0 − Jq0) + γ2[ Dzz(q0) +1

2Dzz(k + q0) ]

dk = Dzz(k)

ν =ω

σ(12.79)

If we eliminate G+−(q + q0, ω) and G−−(q + q0, ω) the resulting equations forfixed ν are

ν − cqπ

= (ν2 − cqcq+q0 +9γ2

4dqdq+q0 )G

+−q

−3γ2

2(dqcq − dq+q0cq+q0 )G

−−q (12.80)

3γ2

2πdq+q0 = (ν2 − cqcq+q0 +

9γ2

4dqdq+q0 )G

−−q −

3γ2

2(dqcq − dq+q0cq+q0 )G

+−q (12.81)

From the condition that the secular determinant of the system of Eqs. (12.80),(12.81) vanish we get two branches for ν2, which we denote as Es(q)

2, withs = ±1:

ν2 = E2s (q) = (cq +

3γ2s

2dq+qo

)(cq+q0 −3γ2s

2dq) (12.82)

The r. h. s. can be written as the product:

Es(q)2 = 4

[

Jq0 − Jq+q0 −γ2

2dq0 − γ2

(

1 + 3s

2dq+qo

) ]

×[

Jq0 − Jq −γ2

2dq0 − γ2(

1 − 3s

2dq)

]

(12.83)

and also as the difference of two squares:

Es(q)2 = (As

q)2 − (Bs

q)2 (12.84)

where

Asq = 2Jq0 − Jq+q0 − Jq − γ2dq0 −

γ2

2

(

1 − 3s

2

)

dq

−γ2

2

(

1 + 3s

2

)

dq+q0

Bsq = Jq − Jq+q0 −

γ2

2

(

1 + 3s

2

)

dq+q0

+γ2

2

(

1 − 3s

2

)

dq (12.85)

12.6. RPA FOR THE AFM CHAIN 293

Since we must have ν2 ≥ 0 there is a stability condition:

(Asq)

2 ≥ (Bsq )2 (12.86)

The dispersion relations have the following properties:

Es(q + q0) = E−s(q) (12.87)

One verifies that

limq→0

E−(q) = 0 ; limq→0

E+(q) > 0

limq→q0

E+(q) = 0

E+(−q0/2) = E+(q0/2) = E−(q0/2) (12.88)

the last equation showing that both branches E± intersect at ±q0/2. This de-generacy is the consequence of the invariance of the AF ground state under thecombination of time reversal and a primitive translation of the atomic lattice,and the result is that one can consider only one branch, with a continuous dis-persion relation in the extended (atomic) BZ [−π/a, π/a . [17]

Exercise 12.7Verify that if q0 = 0 we get the same dispersion relation as in Chap. 6 for theFM chain. Notice that this case corresponds to the FM polarization perpen-dicular to the chain which is unstable for the dipolar forces.

For small q, the coefficient dq has a logarithmic divergence, as we already foundupon studying the FM chain:

dq = a3Dzz(q) = d0 − c2q2 + q2 log

q2

4π+ O(q4) (12.89)

where d0 = 2ζ(3); c2 = 0.666 [17]. This entails a logarithmic term in the dis-persion relation of the acoustic mode. In summary, in the exchange-dominatedregime, with magnetization perpendicular to the chain, we find an acoustic(Goldstone) mode with a logarithmic singularity in the second derivative, andand optical mode, which is obtained by folding the accoustic branch at q =±q0/2. Both are degenerate at the magnetic zone border.

Exercise 12.8Explain why we find a Goldstone mode in this case.

Let us simplify the notation: for any function f(q) we call f(q) = f , andf(q + q0) = f ′. We complete now the calculation of the Green’s function:

G+−(q, ν) =R(ν)/π

(ν2 −E2+)(ν2 −E2

−)(12.90)


where

R(ν) = (ν − c)(ν2 − cc′ +9γ2

4dd′) − 9γ2

4(d′c′ − dc)d′ (12.91)

Notice thatE2

+ −E2− = 3γ2(dc− d′c′) (12.92)

We can now obtain the sublattice magnetization. Let us limit ourselves forsimplicity to the case S = 1/2. As we have done before, we calculate

Φ = − 1

Nσ

∑

k

∫

dωIm G+−(q, ω + iε)N(ω) (12.93)

where N(ω) is the Bose-Einstein distribution function, and for S = 1/2

σ =1/2

1 + 2Φ(12.94)

The sum over k in (12.93) is to be performed inside the magnetic BZ, −q0/2 <q ≤ q0/2, which is half the atomic one. The k summation can be simplified byexploiting relations (12.88), and we can integrate over only one of the branchesby extending the integral to the whole atomic BZ, obtainig as final result theexpression

Φ =1

2N

∑

q

′ A−q

E−(q)− 1 +

1

N

∑

q

′A−

q

E−(q)N(σE−(q)) (12.95)

and the prime in the sum above is there to remind us that we integrate over theatomic BZ.


The first term on the r. h. s. of Eq. (12.95) is the contribution to the magneti-zation deviation from the quantum zero-point fluctuations. The second term isthe thermal contribution.It turns out that the zero-point fluctuations diverge:

Exercise 12.10Prove that the zero point fluctuations diverge due to the singular behaviour ofE−

q at q = 0.

The conclusion is that there is no LRO in the AFM chain, even at T = 0,in the exchange dominated regime.

12.6.2 Dipolar dominated regime

Now all spins are parallel to the chain. Then we are back in the FM case, withonly a change of sign of J , and in our present notation the dispersion relation,

12.7. DIPOLAR INTERACTION IN LAYERS 295

Eq. (12.37) reads

Eq = −4 | J | (1 − cos q) − γ2(d0 +dq

2) (12.96)

We have d0 = −4ζ(3)/a3 and dq0 = 3ζ(3)/a3. We obtain the values

E0 = 6γ2ζ(3)/a3

Eq0 = −8 | J | +5

2a3γ2ζ(3) (12.97)

Due to the competition between exchange and dipolar forces, the magnon nearthe zone border becomes soft. Eq. (12.97) shows that Eq0 = 0 for Ed = Ec

d.This instability leads to the perpendicular (dipole dominated) phase forEd > Ec

d

that we have found before. For Ed > Ecd there is a finite gap in the spectrum,

which is in tune with the fact that the choice of the Weiss ground state breaksa discrete symmetry in this case. There are no zero point fluctuations, as wehave seen, in the FM configuration.

Exercise 12.11Prove that in the dipolar dominated regime Φ converges as long as σ 6= 0.

Therefore in the dipolar regime we have LRO at T = 0, which, if Dyson’sconjecture were right would disappear at any finite T (see Sec. 9.5).

12.7 Dipolar interaction in layers

Let us return to Eqs. (6.16) and (6.17) and suppose we consider a system with2d translational symmetry. We can think of the system as consisting of a fam-ily, finite or infinite, of parallel crystal planes, like a multilayer system with twosurfaces, a semi-infinite one with one surface, or a superlattice, consisting of aninfinite periodic system with a repetition “super-cell” in the direction perpen-dicular to the family of planes. Then we can Fourier transform Eq. (6.16) withrespect to time and to the position vectors of points on each plane, keeping anindex which denotes the plane, since there is no translational symmetry in thedirection perpendicular to the planes, except in the super-lattice case, which weshall exclude. The usual retarded Green’s functions will be written now as

〈〈S+l l‖

;S−n n‖

〉〉 ≡ G+−(

l l‖, n n‖; (t− t0))

(12.98)

where we explicitly separate the plane index from the two-component positionvectors of sites on the corresponding plane, and the index “r” for “retarded” isimplicit. We define the 2d Fourier transform of G:

G+−ln (k‖, ω + iε) =

∑

R‖

eik‖·R‖G+−ln (R‖, ω + iε) (12.99)


where the 2d translational symmetry of the system has been taken into account.We also Fourier transform J :

Jln(k‖) =∑

R‖

eik‖·R‖Jln(R‖) (12.100)

Next step is to apply the RPA to the layered system. At this point we shallmake the simplifying assumption that on each plane n the average 〈Sz

n,R‖〉 is

translationally invariant on the plane, so that it becomes independent on R‖.Then the Green’s functions for fixed k‖, ω are matrices with indices denoting theplanes, which satisfy a system of linear equations. Non-linearity enters throughthe self-consistent calculation of the statistical averages 〈Sz

n〉. This leads to ageneralization of the RPA for layered systems which we might call the layerRPA (LRPA) [18, 19].

We apply now this program to the exchange-dipolar Hamiltonian of Chap. 6.We consider the quantization axis z to be along the plane, for simplicity, in orderto avoid depolarization effects. Then the axes x, z are parallel to the planes,and y is perpendicular to them. We shall also partially Fourier transform thedipolar tensor elements. We define the coefficients

Alm(k‖) =∑

R‖

eik‖·R‖1

(R2‖ + y2

lm)3/21 −

3z2‖

R2‖ + y2

lm

(12.101)

and

B±lm(k‖) =

∑

R‖

eik‖·R‖(xlm ± iylm)2

(R2‖ + y2

lm)5/2(12.102)

Let us simplify the notation and call for any F : Fln(k‖ = 0) ≡ Fln(0). Weobtain the following equations for the Green’s functions:

ω − γB − 2∑

j

(

Jlj(0) − γ2Alj(0))

〈Szj 〉

G+−lm (k‖, ω + iε)

+ 〈Szl 〉∑

j

2Jlj(k‖) +γ2

2Alj(k‖) G+−

jm (k‖, ω + iε)

+3γ2

2〈Sz

l 〉∑

j

B∗lj(k‖)G

−−jm (k‖, ω + iε)

=〈Sz

l 〉π

(12.103)

and

ω + γB + 2∑

j

(

Jlj(0) − γ2Alj(0))

〈Szj 〉

G−−lm (k‖, ω + iε)


−〈Szl 〉∑

j

2Jlj(k‖) +γ2

2Alj(k‖) G−−

jm (k‖, ω + iε)

−3γ2

2〈Sz

l 〉∑

j

Blj(k‖)G+−jm (k‖, ω + iε) = 0 (12.104)

Exercise 12.12Obtain Eqs. (12.103) and (12.104).

Let us now specialize to the case in which each plane is a square 2d lattice,and assume all planes register with each other, so that the resulting 3d latticehas tetragonal symmetry. We consider first n.n. interactions only so that

Jll(k‖) = J∑

δ‖

eiδ‖·k‖

andJl,l±1(k‖) = J = Jl,l±1(k‖ = 0)

12.7.1 Monolayer

It was shown [20, 21] that this model exhibits LRO at finite T and accordinglyit has a phase transition. The current interest in magnetic multilayer systems[22, 23, 24] in view of technical applications arises from the present day capabil-ity of synthezising multilayers with widths varying from several hundred layersdown to a monolayer. If a system is confined to a finite width in the directionperpendicular to the layers, one can still apply Mermin-Wagner theorem unlessthe forces have long range and/or are anisotropic, which is of course the casewith dipolar forces.

For the monolayer there is only one order parameter 〈Sz〉 = σ. We rewriteEqs. (12.103) and (12.104), eliminating repeated indices, as

[

ω − γB

−σ 2 ( J(0) − J(k) ) − γ2

(

A(0) +1

2A(k)

) ]

G+−(k, ω)

+3γ2

2σB∗(k)G−−(k, ω) =

σ

π(12.105)

[

ω + γB

+σ 2 ( J(0) − J(k) ) − γ2

(

A(0) +1

2A(k)

) ]

G−−(k, ω)

−3γ2

2σB(k)G+−(k, ω) = 0 (12.106)


We define

b(k) = a3B(k) ; d(k) = a3Dzz(k) = a3A(k) ; ν = ω/J

andλ(k) = σ 2z (1 − γ(k)) −Ed (d(0) + d(k)/2)

where z = number of first n.n. and γ(k) is the usual structure factor. Thevanishing of the secular determinant of Eqs. (12.105) and (12.106) leads to theeigenvalues

ν2 = λ(k)2 − 9

4| b(k) |2 E2

dσ2 (12.107)

We must require that the r. h. s. of (12.107) be ≥ 0 for stability.For a system magnetized along the z axis on the x, z plane with y perpen-

dicular to the plane, the coefficient B 6= 0, and there is coupling of G+− andG−−. The calculation of A(k) and B(k) [20, 21] can be performed numericallyvery efficiently with Ewald’s method specialized to 2d. We quote now the mainresults.

Let us call φ the angle that q makes with the magnetization (i.e with the zaxis). In the square lattice the dispersion relation for small q is:

(

ν

σEd

)2

= (608.0 sin2 φ+ 73.0 cos2 φ) | q | (12.108)

The approximate calculation in the continuum limit yields only a term in sin2

[21].For a layered finite or semi-infinite system the calculation of the dipolar coef-

ficients can also be done using Ewald’s method. We shall discuss the results forq = 0 near a surface in Chap. 13. Let us briefly mention the results obtained forthe simple case of a bilayer, which corresponds to two magnetic layers separatedby a few layers of a non-magnetic material.

12.7.2 Bilayer

We consider a system of spins localized on the sites of two identical infinitesquare lattices of constant a, parallel to the (z, x) plane and separated by adistance y. Both lattices will be assumed exactly registered on one another(atop geometry), corresponding to a 3d tetragonal structure, and in particulara (010) simple cubic stacking for y = a. We assume first n .n. ferromagneticexchange interactions, both in and between the planes,

Jij =

J‖ in planeJ⊥ interplane

(12.109)

We choose as before the z axis along the magnetization, in the (z, x) plane, andassume ferromagnetic order, both intra- and inter-planar. If all magnetic sites ofthe system are equivalent (symmetric bilayer) then 〈Sz

l 〉 ≡ σ is independent on


the plane index l = (1, 2), We find the inhomogeneous linear system of equationsfor the Green’s functions [25]:

ΩG = Σ (12.110)

Here, we have:

G =

G+−11 (k‖, ω) G+−

12 (k‖, ω)G+−

21 (k‖, ω) G+−22 (k‖, ω)

G−−11 (k‖, ω) G−−

12 (k‖, ω)G−−

21 (k‖, ω) G−−22 (k‖, ω)

(12.111)

Σ =1

π

σ 00 σ0 00 0

(12.112)

The subindices (1,2) in (12.111) refer to the different planes. The matrix Ω is:

Ω =

ω + λσ ση σb∗1 σb∗2ση ω + λσ σb∗2 σb∗1

−σb1 −σb2 ω − λσ −ση−σb2 −σb1 ση ω − λσ

(12.113)

where

λ = −2

[

J11(0) + J12(0) − J11(k‖)

]

+γ2

2

[

2(A11(0) +A12(0)) +A11(k‖)

]

(12.114)

η = 2J12(k‖) +γ2

2A12(k‖)

b∗1 =3γ2

2B∗

11(k‖)

b1 =3γ2

2B11(k‖)

b∗2 =3γ2

2B∗

12(k‖)

b2 =3γ2

2B12(k‖) (12.115)

The Fourier transforms in (12.114) and (12.115) are:

Alj(k‖) =∑

R‖

eik‖·R‖

(R2‖ + Y 2

lj)5/2

[

(R2‖ + Y 2

lj) − 3Z2lj

]

(12.116)

Blj(k‖) =∑

R‖

(R+‖ )2eik‖·R‖

(R2‖ + Y 2

lj)5/2

(12.117)


Jlj(k‖) =∑

δ

eik‖·δ (12.118)

where indices (l, j) identify the planes, and δ is the star of first nearest neigh-bours of a given site.

The average 〈Szl 〉 = σ is related as usual to the spectral distribution function

of the local propagator, an for S = 1/2,

σ =1/2

1 + 2Φ(T )(12.119)

where:

Φ(T ) = − 1

πlim

ε→0+

∫

dω( a

2π

)2∫

d2k‖Im

[

Ω−1(k‖, ω + iε)11]

eβω − 1(12.120)

and of course we can change index 1 by 2, since both planes are equivalent. Thepoles of G, or correspondingly, the zeroes of detΩ(k‖, ω), give the dispersionrelations of the acoustic and optical branches of the magnon spectrum. We find:

D(k‖, ω) = detΩ = ω4 − 2ω2(a20 + a2

1 − b20 − b21) + ∆ (12.121)

with∆ = (a2

0 + a21 − b20 − b21)

2 − 4(b0b1 − a0a1)2 (12.122)

The coefficients in (200) and (201) are defined as:

a0 = σλ , a1 = ση

a1 = ση , b0 = σb1

b0 = σb1 , b1 = σb2 (12.123)

The residues of G at its poles are determined by the cofactor of (Ω(k‖, ω))11. Inthe neighbourhood of T = 0, we calculate the zero-point spin deviation, definedas:

δ(0) =1

2− σ(0) =

Φ(0)

1 + 2Φ(0)(12.124)

where:

Φ(0) = limT→0

Φ = − a

2π

2∫

d2k‖

2∑

α=1

C11(− | ωα |)D′(ωα)

(12.125)

C11(ωα) is the cofactor of (Ω(k‖, ωα))11 at the pole ωα for a fixed k‖, andD′(ωα) is the derivative of detΩ at the pole. The index α denotes the magnonbranch (α = +,−). Both the numerator and the denominator inside the sum in(12.125) are polynomials in ωα of the same order, so it is convenient to introducethe variable

να = ωα/σ (12.126)


to eliminate any explicit dependence on σ. We call

G(να) ≡ C11(−ωα)

D′(ωα)(12.127)

and get for the inverse critical temperature the equation

J‖

kBTc=

2

N

∑

k‖

2∑

α=1

G(να)

να(12.128)

The numerical results can be fitted within 5% with the formula [25]

J‖

kBTc= a+ b| lnEd| (12.129)

The coefficients a and b depend on the particular parameters and geometry.The logarithmic divergence as Ed → 0 in Eq. (12.129) arises from the integralover 2d k‖ space in Eq. (12.128) which determines Tc. The vanishing of Tc inthe absence of dipolar interactions is consistent with Mermin-Wagner theorem.

The spin deviation at T = 0, which measures the effect of the zero pointspin fluctuations can be numerically calculated as a function of Ed for 0 < Ed ≤2 × 10−2 and it turns out to be linear in Ed [25] for small Ed.

From the point of view of the existence of LRO the interesting branch is theacoustic one. The important result is:

Exercise 12.13Prove that for k‖ = 0, A12(0) = −B12(0) and that this implies, that

limk‖→0

ν−(k‖) = 0 (12.130)

That is, the magnon spectrum has no gap. The dipolar forces stabilize the LROexclusively because they change the functional form of the dispersion relation forsmall q. If the interplanar terms B12, A12 and J12 are neglected, correspondingto the limit of infinite distance between the planes, then ν−(k‖) is proportional

to√

|k‖| for very small |k‖|. This is the dispersion relation of a monolayer wefound before.

For the coupled planes, instead, ν2−(k‖) is quadratic in |k‖|. If we study

ν2−(kz), as a function of Ed, we find that the second derivative at kz = 0 in-

creases with Ed (it vanishes for Ed = 0), up to a maximum and then decreases,becoming eventually negative for Ed > Ec

d which depends on the parameters.This implies that ν− is pure imaginary for Ed > Ec

d. This instability is a con-sequence of the static (small k) dipolar field, which favours AFM alignment ofthe parallel planes, for the chosen geometry. As Ed grows, the stability of theFM order in 3d depends on the competition between interplane exchange anddipolar interactions. Although Ed is usually very small, at long distances, orsmall |k‖|, the dipolar interaction dominates the dispersion relations, and this


is why the 3d FM inter-planar alignment is not stable in the s.c. structure [26].Each plane will still be FM, but the relative orientation of the planes will not.

In conclusion, as Ed grows from 0 one first finds the stabilization of LRO,but when Ed reaches a critical value which depends on the system parameters,a crossover is observed from 2d to 3d behaviour, which for this geometry desta-bilizes the FM pairing of the planes.

Since the exchange interaction is clearly dominant for large k ≈ π/a, weknow that ν2 must become positive as k grows. That means that if we are inthe instability regime (Ed ≥ Ec

d) there must be some k = kc such that ν2 ≥ 0for k ≥ kc. This defines a length scale k−1

c which could be the typical size ofdomains. Inside one of these the bilayer might have a FM alignment of bothplanes. This is one possible configuration that may compete to minimize thefree energy of the system.

References

1. Mermin, N. D. and Wagner, H. (1966) Phys. Rev. Lett. 17, 1133.

2. Hohenberg, P. C.(1967), Phys. Rev. 158, 383.

3. Majlis, N., Selzer, S. and Strinati, G. C. (1992) Phys. Rev. 45, 7872;(1993) Phys. Rev. 48, 957.

4. Birgenau, R. J., Guggenheim, H. J. and Shirane, G. (1969) Phys. Rev.Lett. 22, 720.

5. Tinus, A. M. C., de Jonge, W. J. M. and Kopinga, K. (1985) Phys. Rev.B32, 3154.

6. Dingle, R., Lines, M .E. and Holt, S. L. (1969), Phys. Rev. 187, 643.

7. Bogoliubov, N.N. (1962) Physik. Abhandl. Sowietunion 6, 1,113,229.

8. Dyson, Freeman J. (1969) Commun. Math. Phys. 12, 91.

9. Ewald, P. P. (1921) Ann. Phys. (Leipz.) 64, 253.

10. Born, Max and Huang, Kun (1956) Dynamical Theory of Crystal Lattices,Oxford at the Clarendon Press, London.

11. Abramowitz, Milton and Stegun, Irene A. (eds.) (1965) Handbook ofMathematical Functions Dover Publications, Inc., New York.

12. Peierls, R. E. (1936) Proc. Cambridge Phil. Soc. 32, 477.

13. Gallavoti, G. and Miracle-Sole, S. (1967) Commun. Math. Phys. 5, 317.

14. Rushbrooke, G. and Ursell, H. (1948) Proc. Cambridge Phil. Soc. 44,263.


15. Ruelle, D. (1968) Commun. Math. Phys. 9, 267.

16. Kac, M. and Thompson, C. J. (1968) Critical behaviour of several latticemodels with long range interaction, Preprint, Rockefeller University.

17. Humel, M., Pich, C. and Schwabl, F. (1997) Preprint, cond-mat/9703218,25th March.

18. Selzer, S. and Majlis, N. (1980) J. Mag. and Mag. Mat. 15-18, 1095.

19. The-Hung, Diep, Levy, J. C. S. and Nagai, O. (1979) Phys. Stat. Sol. (b)93, 351.

20. Maleev, S. V. (1976) Sov. Phys. JETP 43, 1240.

21. Yafet, Y., Kwo J. and Gyorgy, E. M. (1988) Phys.Rev. B33, 6519.

22. Parkin, S. S. P. (1991) Phys. Rev. Lett. 67, 3598.

23. Celotta, R. J. and Pierce, D. T. (1986) Science 234, 249.

24. Baibich M. N. et al. (1988) Phys. Rev. Lett. 69, 969.

25. de Arruda, A. S., Majlis, N., Selzer, S. and Figueiredo, W. (1955) Phys.Rev. B51, 3933.

26. Cohen, M. H. and Keffer, F. (1955) Phys. Rev. 99, 1128;1135.

Chapter 13

Surface Magnetism

13.1 Introduction

The study of magnetic phenomena at surfaces has been greatly stimulated bythe availability of several techniques which are capable to probe the first layersof a magnetic sample. Among them, the progress in methods for ultra highvacuum (UHV) deposition of epitaxial films , in particular molecular beam epi-taxy (MBE) [1]. Other methods, as reflection high energy electron diffraction(RHEED), low energy electron diffraction (LEED) and Auger electron spec-troscopy (AES) [2], allow us to determine the detailed atomic structure of thesynthesized films, one of the critical problems in this field. Finally, the studyof the surface magnetization is facilitated by the application of several tech-niques like spin polarized low energy electron diffraction (SPLEED), electroncapture spectroscopy (ECS) [3], spin-analyzed photoemission, electron scanningspectroscopy with spin polarization analysis (SEMPA) [2] and the atomic forcemicroscope (AFM) [4].

For insulators, an ordered surface at temperatures at which the bulk is dis-ordered can only be expected as the result of anisotropy or long range of theinteractions. By definition, an ordered surface phase extends only through afinite length into the bulk. Therefore we require LRO in a two dimensional,finite width, surface layer. According to the Mermin-Wagner theorem LRO ina 2d system cannot exist unless the interactions are anisotropic and/or havelong range parallel to the surface. The theorem was in fact proven only for astrictly 2d system, but it is not difficult to realize that it can be extended toa finite number of layers. To see this, one can imagine that a perpendicularinter-plane coupling is turned on between a series of parallel two dimensionallayers. Independently of the nature of the interactions, as long as they are uni-form in the directions parallel to the surface, they cannot destroy the LRO oneach plane, although they can lead to an infinite variety of relative orientationsof the magnetization on the different planes. Therefore we conjecture that thesame conditions for the establishment of LRO as in a strictly 2d system willapply.

305

306 CHAPTER 13. SURFACE MAGNETISM

We remark that in order to have surface magnetism at temperatures greaterthan the bulk transition temperature, the surface layer must not only be or-dered, but its own critical temperature must be greater than the bulk one [5].Neel [6] analyzed the possibility of surface anisotropy in a FM due to localchanges in magneto-elastic and magneto-crystalline interactions at the surface.Similar arguments based on the difference in local environments between sur-face and bulk spins suggest that one can also expect variations of the exchangeanisotropy in going from bulk to surface.

Surface magnetism is found in the rare earths Gd [7] and Tb [8]. Thesematerials are metallic, but their magnetic behaviour is very well described bythe localized spins of incomplete f ionic shells, which interact through indirectexchange of the RKKY type.

13.2 MFA treatment of surfaces

In the presence of a surface, the translation symmetry group is two dimensional.We can describe the semi-infinite lattice as a family of crystal planes parallelto the surface. We choose a pair of primitive translations on each plane, whichfor simplicity we assume identical on all planes, although this depends on theparticular crystal structure. Let us call n1, n2 the pair of integer coordinates ofa site on a particular plane, and denote by n3 the coordinate of the plane. Wechoose n3 = 0 at the surface and n3 > 0 for the bulk. In Fig. 13.1 we show asemi-infinite s.c. FM lattice so described.

Let us assume that the translation symmetry along the planes applies aswell to 〈Sz

n,n3〉, so that this quantity only depends on n3:

〈Szn,n3

〉 = σn3 (13.1)

It is now straightforward to obtain the molecular field equations. This case isin principle similar to helimagnetism, in that we have many different planarsublattices, each of which has a given σn3 . The molecular field equations are

σn3 = BS(γβSBmoln3

) , ∀ n3 ≥ 0

Bmoln3

=∑

µ=0,±1

σn3+µzµJn3,n3+µ (13.2)

where zµ is the corresponding number of equidistant neighbours. Let us look atsome simple examples to clarify the notation used in Eq. (13.2).

Simple cubic (s.c.) (100) surface

We take an exchange interaction of first n. n. range, with value J in the bulk,J‖ for bonds on the surface plane, and J⊥ for bonds between planes n3 = 0 (sur-face) and n3 = 1, as depicted in Fig. 13.1. In the absence of an external fieldand of anisotropy the orientation of the magnetization is immaterial. However,

13.2. MFA TREATMENT OF SURFACES 307

Figure 13.1: Schematic description of the surface of a semi-infinite s.c. fer-romagnet. Horizontal axes n1, n2. The vertical coordinate n3 increases intothe bulk. Different values of Jij in the surface region are indicated. The bulkexchange constant is denoted J .

in real systems, the presence of the dipolar interactions, in the absence of otheranisotropies, orients the magnetization parallel to the surface. We assume thisis the case, which results in a vanishing depolarizing field in the bulk. We sallneglect, for simplicity, the possibility of of a surface depolarizing field.

The molecular field on the different planes is:

Bmol0 = 4J‖σ0 + J⊥σ1

Bmol1 = 4Jσ1 + J⊥σ0 + Jσ2

Bmoln3

= J(σn3+1 + 4σn3 + σn3−1) , ∀ n3 > 1. (13.3)


Body centred cubic (b.c.c.) (100) surface

In this case spins on the (100) planes are second nearest neighbours of eachother, so that the corresponding coupling constant vanishes. We have:

Bmol0 = 4J⊥σ1

Bmol1 = 4Jσ2 + 4J⊥σ0

Bmoln3

= 4J(σn3+1 + σn3−1) , n3 > 1 (13.4)

Exercise 13.1Obtain the expressions of the molecular fields for the (111) face of the f.c.c.lattice under the previous approximations. In this case we assume the magneti-zation is along (110).

At T = 0, if all the interactions are FM, we expect to find the system inthe Weiss ground state, irrespective of the values of the parameters. At finite Twe must solve the set of equations (13.2) self-consistently. One can pose oneselfthe questions (in the following s stands for surface, b for bulk):

1. Does σs vanish at T < T bc ?

2. Is it possible that σs > σb ? And, if so,

3. Is it possible that the surface stays magnetized at T > T bc with the bulk

in the PM phase (σb = 0)?

Within MFA the answer is yes to 2) and 3), no to 1) [9]. However, calculationswhich take into account thermal fluctuations of the transverse spin components,even within the FSWA or RPA yield different answers.

Let us start with 1). For n3 large enough, the molecular field equations areidentical for all successive planes and then we must get the same σb as withouta surface and the same critical temperature T b

c . At T = 0 in a FM we knowthat σn3 = S on all planes, so that the profile is completely flat. At T > 0,in general, σn3 6= σb near the surface. If J‖ > J and/or J⊥ > J we couldhave σs > σb. In this case MFA can yield a non-zero surface magnetization atT > T b

c . The magnetization decreases as n3 grows and it vanishes deep insidethe crystal. As already mentioned, this result is a consequence of the MFA, andwe shall see that the picture can be different in the RPA. At any rate, whenthe surface remains ordered at T > T b

c , the MFA predicts a second transition,called the surface transition, at some T s

c at which σs → 0. In the case thatthe transition to the PM phase at the surface occurs at T s

c < T bc we speak of

the ordinary surface transition, and in this case the surface is controlled by thebulk. When both temperatures coincide the transition is multicritical and iscalled special. When there is a surface transition, so that T s

c > T bc , the MFA

predicts a discontinuity in the first derivative of the magnetization with respectto T , and this is called the extraordinary transition.

13.3. SURFACE EXCITATIONS 309

The critical behaviour of magnetic surfaces has been analyzed in a seriesof papers [9, 10]. We shall proceed now to describe the results obtained withthe self consistent LRPA (Layer RPA ) for ferromagnets, which was alreadyintroduced in the previous chapter [12].

13.3 Surface excitations

We extend now the LRPA by considering as many sublattices as necessary todescribe reasonably the non uniformity of the magnetization near to a surface.We assume that the local average 〈 σz(n‖, n) 〉 = σz

n is uniform across eachplane n, where n stands for n3, so that n = 0 is the surface plane and n ≥ 0for the semi-infinite lattice. The argument that follows can be applied at low Tto obtain a spin-wave theory of surface magnons. If however the magnetizationis not saturated and we take it as a T dependent parameter, we shall need toextend the treatment and solve the self-consistency problem by application ofthe LRPA.

The translations perpendicular to the surface are not symmetry operationsof the Hamiltonian. As already mentioned, those parallel to the surface canstill be considered as symmetry operations if the system extends to infinity inboth directions parallel to the surface plane, which we will assume throughout.If Bloch waves are used as a basis to expand the magnon wave functions, thek⊥ component of the wave vector need not be real as in an infinite system.A Bloch wave with a real k⊥ > 0 arriving at the surface from the bulk willbe reflected, with a perpendicular wave-vector −k⊥, and in general with somepossible change of phase and amplitude, so that as a result we expect that bulksolutions are standing waves, with a continuum spectrum which resembles veryclosely that of the infinite bulk.

On the other hand, since the system is bounded by the surface, we can con-sider stationary wave functions with Im(k⊥) 6= 0, which must vanish far intothe bulk, since they must be normalizable over the whole semi-infinite spaceoccupied by the system. These waves, whose amplitude decreases exponentiallytowards the bulk, are the basis for obtaining surface-localized magnons. There-fore we are again facing the problem of the analytic continuation of the bulkspectrum as in the previous discussion of indirect exchange (see Chap. 10). Byimposing the correct boundary conditions at the surface and the reality of theenergy of the magnon energy we can in fact obtain in some cases the surfacemagnon energy and wave function [13].

These general considerations are valid whatever the Hamiltonian we adoptas model for the system. In most of this chapter we shall leave aside, for simplic-ity, the dipolar interactions. In a macroscopic approach, the exchange-dipolarsurface magnons were studied by Damon and Eshbach [11]. A microscopic cal-culation of dipolar-exchange spin excitations in a system with a surface, at finiteT , is cumbersome but it does not present new complications, although we arenot aware of any example in the literature.


13.4 LRPA method

Let us consider again the isotropic Heisenberg hamiltonian

H = −∑

l6=m

JlmSl · Sm (13.5)

and the retarded Green’s functions

〈〈S+l l‖

;S−n n‖

〉〉(r) ≡ G(r)(

l l‖, n n‖; (t− t0))

(13.6)

with the Fourier transform

G(r)ln (k‖, ω + iε) =

∑

R‖

eik‖·R‖G(r)ln (R‖, ω + iε) (13.7)

with R‖ = l‖ − n‖, using the translation invariance. We can also Fouriertransform the exchange coupling:

Jln(k‖) =∑

R‖

eik‖·R‖Jln(R‖) (13.8)

We write the equation of motion for the retarded Green’s function of Eq. (13.6)and make the usual factorization of the extra factor Sz in the Green’s functionwith three operators. As in the MFA, we assume that in the present non-uniformcase the average 〈 Sz

n 〉 is a function of the plane index. The resulting equationsfor the Green’s functions are conveniently written in dimensionless form, bydefining:

ν = ω/2J ; gln = JGln ; εln = Jln/J (13.9)

and we find that the matrix g (with plane indices l, n) satisfies the equation:

(ν1 − heff )g = Σ/π (13.10)

where

(Σ)lj = δljσl , σl = 〈 Szl 〉

(heff )lj = δlj∑

i

εij(k‖ = 0)σi − σlεjl(k‖) (13.11)

The matrix equation (13.10) can be written

Ωg =1

πΣ (13.12)

For planes well inside the bulk, where there are no alterations of the exchangeconstants or of the local coordination, the diagonal matrix element of heff is

heffmm = heff

b ≡ z⊥σb + z‖σb( 1 − γ0(k‖)) (13.13)

13.4. LRPA METHOD 311

Here γ0(k‖) and z‖ are respectively the structure factor and the coordinationnumber of the two-dimensional planar lattice, z⊥ is the number of first n.n. ofa given site which belong to adjacent layers and σb is the bulk magnetization.All diagonal matrix elements (13.13) are equal for large enough m. Thereforewe shall write the diagonal term for all m as the bulk one (Eq. (13.13)) plus alocal deviation and define

ν − (heff )mm ≡ 2t+ αmm (13.14)

where2t ≡ ν − heff

b , αmm ≡ heffb − heff

mm (13.15)

We rewrite accordingly the elements of Ω as

Ωg =1

πΣ (13.16)

where

Ωll = 2t+ αll

Ωlj ≡ βlj = −σlεjl(k‖) , j 6= l . (13.17)

From Eq. (13.10) we get:

g =1

πΩ−1Σ (13.18)

so that the main mathematical problem is now the inversion of Ω.The extension to this case of the fluctuation-dissipation theorem and of

Callen’s formulas for the self-consistent evaluation of σn (see Chap. 6) is im-mediate [12]. In Eq. (5.90) we express the local magnetization in terms of onefunction Ψ in a uniform FM, while in the non-uniform case we need one suchfunction for each plane, so that we have, for plane n:

Ψn = − 1

Ns

∑

k‖

∫

dω

πIm(Ω−1)nn(k‖, ω + iε)N(ω) (13.19)

For a FM with spin S the layer magetization is given by Callen’s formula forthe layer:

〈 Szn 〉 =

(S − Ψn)(1 + Ψn)2S+1 + (S + 1 + Ψn)Ψ2S+1n

(1 + Ψn)2S+1 − Ψ2S+1n

(13.20)

Let us now look at the (100) surface of a s.c. latice as a specific example. Inprinciple there is an indeterminate number of different values of σn if the non-uniform profile extends to infinity. However, it is reasonable to assume thatthe influence of the surface only affects a finite number of planes. Althoughthis may seem dubious for the dipolar interactions, one can verify that theircontribution to the effective hamiltonian heff decays very rapidly, in fact expo-nentially fast, towards the interior of the system [14]. Therefore we make thefollowing simplifying assumptions:


1. The coupling constants are different from the bulk only for intra-surfacebonds and bonds between surface and the next plane;

2. The magnetization profile saturates to the bulk value at the third planeof the semi-infinite system, i.e. σn = σb , n ≥ 2 .

This situation is indicated in Fig. 13.1. For this case one can easily work outthe matrix elements of Ω:

α00x = 2σb − σ1ε⊥ + (σb − σ0)Λk

α11 = σb − σ0ε⊥ + (σb − σ1)Λk

α22 = (σb − σ1)

β01 = σ0ε⊥

β10 = σ0ε⊥

β12 = σ1

β21 = 1 (13.21)

where we definedΛk = 4

(

1 − γ(k‖))

(13.22)

All β’s following the ones above are = 1.The matrices Ω and Σ are:

Ω =

2t+ α00 β01 0 0 0 · · ·β10 2t+ α11 β12 0 0 · · ·0 β21 2t+ α22 1 0 · · ·0 0 1 2t 1 · · ·...

......

. . .. . .

. . .

(13.23)

Σ =1

π

σ0 0 0 0 · · ·0 σ1 0 0 · · ·0 0 σb 0 · · ·0 0 0 σb 0...

......

.... . .

(13.24)

The three-diagonal matrix (13.23) can be directly inverted:

(Ω−1)nm =Adj(Ωnm)

detΩ(13.25)

Let us for the moment admit that we have a finite number of planes in thesystem, say N . Then Ω is N × N . Let us call DN−p , 0 ≤ p ≤ N − 1,the determinants of the lower minor of Ω of order successively decreasing withincreasing p, so that DN = detΩ. In particular, if we want to calculate thesurface magnetization we need

(Ω−1)00 =DN−1

DN(13.26)


The direct expansion of DN by the first row gives

DN = (2t+ α00)DN−1 − β10β01DN−2 (13.27)

For p ≥ 3, in our special example, the recursive relations between these deter-minants have coefficients independent on p :

DN−p = 2tDN−p−1 −DN−p−2 (13.28)

Suppose that in the interval of t we shall be interested in,DN−p 6= 0 , ∀ p ≥ p0 for some integer p0. Then for p > p0 + 1 let us divideEq. (13.28) by DN−p−1:

DN−p

DN−p−1= 2t− DN−p−2

DN−p−1(13.29)

We take the thermodynamic limit and define

limN→∞

DN−p−1

DN−p= ξ (13.30)

assuming it exists, in which case it must be obviously p -independent. Weremark that the ratio in Eq. (13.29) can be expressed as a continued fraction:

DN−p−1

DN−p=

1

2t− DN−p−2

DN−p−1

(13.31)

which as N → ∞ can be written as

ξ =1

2t− ξ(13.32)

orξ + ξ−1 = 2t (13.33)

In the more general case in which the quantities in Eq. (13.28) are matrices, ξis called the transfer matrix, and we shall use this name for the case in hand inwhich it is a scalar.

The solutions of Eq. (13.33) are

ξ = t±√

t2 − 1 (13.34)

The continued fraction which results from the repeated iteration of (13.32),

ξ =1

2t− 12t−···

(13.35)

converges for | t |> 1 to the solution of (13.33) which has the minimum absolutevalue, which implies | ξ |< 1 [15]. The other solution is spurious. This iscrucial in searching for the poles of the Green’s function, as we shall see. In


Figure 13.2: Convergence region of the continued fraccion expansion for thetransfer matrix .

Fig. 13.2 we show the convergence circle for the continued fraction (13.32). Inorder to understand the physical meaning of the transfer matrix let us return tothe Green’s function equations (13.10), and write in particular an off-diagonalelement, in which case the r. h. s. vanishes, and we write

Ωn,n−1 gn−1,m + Ωnn gn,m + Ωn,n+1 gn+1,m = 0 (13.36)

For n,m large enough, we would have

gn−1,m + 2tgn,m + gn+1,m = 0 (13.37)

Suppose now that

gn+1,m = ρgn,m (13.38)

If we use this relation in (13.37) we find that ρ satisfies the quadratic equation

ρ2 + 2tρ+ 1 = 0 (13.39)


so that −ρ = ξ. We get further insight into the role played by the transfermatrix if we expand heff in terms of its eigenvectors and eigenvalues, as

heff =∑

µ

Eµ | µ〉〈µ | (13.40)

Then the resolvent operator Ω−1 is

(Ω−1)(ν + iε) =∑

µ

| µ〉〈µ |ν + iε−Eµ

(13.41)

and the generic matrix element of the retarded Green’s function is

gn,m =∑

µ

〈n | µ〉〈µ | m〉ν + iε−Eµ

σm

π(13.42)

where 〈n | µ〉 ≡ ψ(µ)n = amplitude of the wave function of the magnetic exci-

tation with energy Eµ at site n . If we assume now that for fixed m (13.38) isvalid, we must have as well

ψ(µ)n+1 = ρ(µ)ψ(µ)

n (13.43)

With this result in hand, let us suppose that | µ〉 is a surface magnon. Thenwe must require | ρ(µ) |< 1, since we know that the amplitude of a surface wavemust decrease towards the bulk. If on the contrary | µ〉 where a bulk magnon,we require | ρ(µ) |= 1, because a bulk excitation must propagate throughout thewhole system. We can then write for a bulk magnon

ρ = eik⊥a (13.44)

with k⊥ = real. As a matter of fact, we can also express ρ for surface states asin (13.44), but then Imk⊥ > 0.Substituting ρ from Eq. (13.44) into Eq. (13.39) we have

t = − cosk⊥a (13.45)

Suppose we choose for the s.c. lattice the axes x, z along the surface, and k⊥ =ky. With this choice, we can go back to Eqs. (13.13) and (13.15) and we find,for bulk states, that the poles of the Green’s function are

ν = 6 − 2(cos kxa+ cos kya+ cos kza) (13.46)

which is exactly the dispersion relation for bulk magnons in a s.c. ferromagnet(see Chap. 4). We verify that the bulk states have a uni-modular transfermatrix and that their dispersion relation is the same for the semi-infinite as forthe infinite system.


13.5 Wave functions for bulk and surface

The matrix (13.10) has the same form as the tight-binding Hamiltonian for asemi-infinite 1d metal with a first n.n. range hopping term, or, with the changeν → ω2, that of phonons in a chain with short range elastic coupling betweenneighbouring atoms, etc. We must however keep in mind that the matrix ele-ments in (13.10) depend on k‖, which restores the three-dimensional behaviourof the system. We can say that the representation chosen has transformed asemi-infinite 3d system into a family of semi-infinite chains , each one depen-dent parametrically on the corresponding k‖ within the 2d Brillouin zone of theplanar lattice. A given eigenstate of the semi-infinite chain can be labelled byk‖ and some other quantum numbers that we can call generically µ, so thatthe corresponding ket can be denoted as | µk‖〉. Let us simplify the notationand call this state | ψ〉 and the eigenvalue E(µk‖) = ν. The layer index inthis representation can be interpreted as the site index in the one dimensionalmapping of the problem. Let us then write the Schrodinger equation

heff | ψ〉 = ν | ψ〉 (13.47)

and expand the eigenket in terms of site eigenkets | n〉:

| ψ〉 =∑

n

ψn | n〉 (13.48)

Let us now consider the simple case of the s.c. lattice without change of surfaceparameters.

Exercise 13.2Show that if there is no change of parameters at the surface, and if T is verylow, so that the magnetization profile is flat, one finds

α00 = 1 , αnn = 0 , ∀ n > 0 (13.49)

βi,i±1 = 1 , βi,j = 0 otherwise. (13.50)

Notice that this Hamiltonian is equivalent to that of a semi-infinite chain with thediagonal end parameter altered, since in the case without surface perturbationone should have α00 = 0 aswell. Therefore the set of linear equations satisfiedby the amplitudes ψn in (13.47) are:

(2t+ 1)ψ0 + ψ1 = 0

ψ0 + 2tψ1 + ψ2 = 0

· · ·ψn−1 + 2tψn + ψn+1 = 0 (13.51)

We already know that the Ansatz ψn = ρψn−1 solves these equations. Let usfirst consider the bulk states, so that

ρ = eiθ (13.52)

13.6. SURFACE DENSITY OF MAGNON STATES 317

From Eq. (13.39) we also know that ρ−1 = e−iθ is on the same footing as ρ,so that we can construct a general solution of the last of (13.51) by combiningthese two solutions. Since we can iterate the application of the transfer matrixwe finally arrive at the general solution

ψn = A(

einθ + ηeiφe−inθ)

(13.53)

with the coefficient η and the phase φ to be determined by the boundary con-dition provided by the first equation of the system (13.51), which is obviouslynot automatically satisfied by (13.53). The constant A will be fixed by the con-dition that the wave function be normalized to one. From Eq. (13.39) we alsoget t = − cos θ.

Exercise 13.3Show that the solution of the system (13.51) with A = 1 is

ψn = 2e−iθ/2 cos (n+ 1/2)θ (13.54)

13.6 Surface density of magnon states

The density of states of a Hamiltonian H is defined as

ρ(ν) = Trδ(ν −H) (13.55)

In the case of our effective one dimensional Hamiltonian, we can calculate thetrace for a particular k‖, which yields a partial density of states, so that we justneed to sum over all k‖ within the first Brillouin zone (BZ) of the planar latticeto get the total density of states. We can calculate the trace (which we know isinvariant under unitary transformations) in the basis of the layer states, definedas

| n〉 = S−k‖,n | 0〉 (13.56)

where we assume that the Weiss ground state | 0〉 is polarized ↑ and we simplifiedthe notation avoiding the explicit reference to k‖. Then

ρ(ν) =∑

n

∑

µ

〈n | µ〉δ(ν −Eµ)〈µ | n〉 (13.57)

where we have inserted twice the unit operator

∑

µ

| µ〉〈µ | = 1 (13.58)

We can also write (13.57) as

ρ(ν) = limε→0

∓ 1

πIm 1

ν −H ± iε (13.59)


or

ρ(ν) = limε→0

∓ 1

π

∑

n

Im 〈n | 1

ν −H ± iε| n〉 (13.60)

The expression between curly brackets above is precisely

(Ω−1) nn .

From Eq. (13.60) we infer that this quantity is the local projection of the totaldensity of states on that particular layer, for that particular k‖. Let us considerthe previous example of a s.c. (001) surface at very low temperatures, so thatσn = σb , ∀ n=. Let us for a moment consider also the case in which the param-eters at the surface coincide with the bulk ones. This does not mean howeverthat the system is uniform, since near the surface there are less bonds, and thisis equivalent to a perturbation potential, as we have just seen. In this particu-lar example, all the diagonal perturbation terms αnn = 0 with the exception ofα00 = 1.

Exercise 13.4Show that in this case we obtain

(Ω−1)00(ν + iε) =1

2t+ 1 − ξ(13.61)

We just found that for bulk magnons ξ = e−iφ, where we made a convenientchoice of sign in the exponent. We also have t = cosφ from our previous results,so that we find

ρ0(ν) = − 1

πIm (Ω−1)00(ν + iε) =

1

2π

√

1 − t

1 + t(13.62)

The bulk states contribute to the imaginary part of the resolvent operator inthe interval of energies where the continued fraction for ξ has a cut along theinterval −1 ≤ t ≤ 1 (see Fig. 13.2). As a consequence, in this continuum rangeof bulk energies the Green’s function also has a cut on the real axis. On theother hand, for | ξ |< 1 we can find real poles of g00. Each localized eigenstate(surface magnon) adds a delta function contribution to the density of states. Wesee from Eq. (13.61), which corresponds to the special case in which the surfaceparameters are the same as in the bulk, that the resolvent( or equivalently g00)has only one real pole, at ξ = −1, which according to the discussion in theprevious section does not describe a surface state, for which we need | ξ |< 1.

In the general case in which we have different parameters in the surface re-gion, we may find one or more real poles of the Green’s functions, besides thecontinuum of bulk sates. A real pole with | ξ |< 1 determines the energy andthe penetration length λ = −(log | ξ |)−1 of the corresponding surface magnon.

This example indicates how one can construct an algorithm to obtain thesurface states spectrum as a function of k‖, and to calculate the resolvent op-erator both in the bulk continuum and in the discrete sector of the spectrum

13.7. SURFACE PHASE-TRANSITIONS 319

contributed by the surface magnons. With this we can in turn calculate the func-tion Ψn and therefrom the layer magnetization. This program for the simplecubic (010) surface leads to results shown in Fig. 13.3, where we plot ω/(2Jσb)for the different branches of surface magnons, together with the limits of thecontinuum spectrum, vs. Λ = 1 − γ(k‖), where γ is the 2d structure factordefined before. In Fig. 13.4 we show a profile obtained for the case in which

Figure 13.3: Dispersion relation of surface magnons in a s.c. FM. ν =ω/(2Jσb(T )) vs. Λ = 1−γ(k‖). S = 1/2, ε‖ = 1. For T/Tc = 0.17, crosses cor-respond to ε⊥ = 1, open circles to ε⊥ = 2. Full circles correspond to T/Tc = 0.3and ε⊥ = 1. From [16]

the magnetization of the third plane has been fixed at the bulk value. Thetemperature is below the bulk critical temperature. Note that even in the caseε⊥ = ε‖ = 1 the surface magnetization is smaller than the bulk one at finite T ,as expected.

13.7 Surface phase-transitions

LRPA calculations for several lattice structures show that as the coupling J⊥ ofthe surface to the bulk decreases the surface magnetization also decreases. Thistendency can be clearly verified in Fig. 13.5. The possibility that the surface bemagnetized for T > T b

c and accordingly that there be a phase transition localizedat the surface at T s

c > T bc has been studied with a Heisenberg Hamiltonian with

exchange anisotropy [5]. One may conveniently write such a Hamiltonian with


Figure 13.4: Magnetization profile near the surface of a s.c. S = 1/2 FM forε⊥ = ε‖ = 1. We plot σn = 〈Sz

n(T )〉/〈Szb (T )〉 vs. n = plane index. The surface

plane is n = 0. The values of τ = kBT/6J are indicated on the curves, and thecritical value is τc = 0.33. [16]

the restriction to first n.n. interactions as

H = −(JI/η)∑

〈ij〉

(

Si · Sj + (η − 1)Szi S

zj

)

(13.63)

Then when η → 1 we obtain the isotropic Heisenberg Hamiltonian, while ifη → ∞ we obtain the Ising Hamiltonian. For any finite η > 1, calling J = J I/η,we get the bulk spin-wave dispersion relation

ω(k)

2Jzσ= ∆ + (1 − γ(k)) (13.64)

with ∆ = (η − 1)= gap at k = 0. This gap will occur in any dimension of thelattice, so that in this case Mermin-Wagner theorem does not discard LRO atfinite temperatures for d < 3. The LRO however might still be absent in 1d atfinite T due to kinks which increase the entropy, as discussed in Chap. 12. In3d, when η > 1 we find for J > 0 a FM ground state with spins aligned along z(easy axis). If η < 1 instead, spins tend to align on the plane perpendicular toz (easy plane). In 2d, if we take z ⊥ to the lattice plane, we obtain the “XY”model. In this case, at low T 〈 Sz

i 〉 ≈ 0 and one can treat the spins as two-dimensional vector operators. This model has an interesting phase transition as

13.7. SURFACE PHASE-TRANSITIONS 321

Figure 13.5: Bulk σb(T )/S (A) and surface σs(T )/S for a s.c. FM vs. t =kBT/6J for S = 1/2 and ε‖ = 1. The values of ε⊥ for the different curves are:1(B); 0.5(C) and 0.1(D) [16].

T decreases, from the PM disordered phase to a phase which has no long rangeorder but in which the magnetization on the plane has long-range correlations.This phase, described by Kosterlitz and Thouless [17] has topological defects inthe form of vortices of the magnetization field.

In summary, systems with η > 1 belong to the Ising universality class, thosewith η < 1 to the XY universality class, and those with eta = 1 to the Heisen-berg one.

In the anisotropic case with η > 1, or at any rate with ηs > 1 on the surfaceplane, we can expect a surface phase transition. If ηs is large enough there isthe possibility of a surface critical temperature T s

c > T bc . This behaviour is


Figure 13.6: Temperature dependence of the SPLEED asymmetry A, pro-portional to the surface magnetization, in a Gd (0001) film of 140 A thick.Ts = 315 K, Tb = 293 K. No asymmetry is detected on a contaminated surface(open circles). [7]

observed in Gd [7] (see Fig. 13.6) and Tb [8] (see Fig. 13.7). The cause ofthe sudden increase of the surface magnetization in these metals at T near thesurface transition to the PM phase is not clear [18].

An ordinary surface phase transition was observed in the EuS (111) surface[19]. These insulator is a model bulk Heisenberg FM, but the measured criticalexponent of the surface, different from the bulk, could signal the presence ofsurface anisotropy.

Calculations of the magnetization profile within the LRPA yield a surfacephase transition for a sufficiently anisotropic surface, with ηs > ηc [5]. The min-imum surface anisotropy ηc depends on the geometry and on the other surfaceparameters.

13.8. DIPOLAR SURFACE EFFECTS 323

Figure 13.7: Electron spin polarization of the topmost surface layer of 1mmthick Tb samples, as function of T . Tcb is the bulk Curie temperature and TNb

the bulk Neel temperature [3].[8]

13.8 Dipolar surface effects

Let us consider now the effect of dipolar interactions at the surface. Until nowwe have explicitly assumed (see Chapter 7) that all points at which the dipolarfields were calculated were very far from the surface. We shall now considerpoints on or near the surface of a semi-infinite cubic lattice of localized spins.In this case the dipolar sums extend over a set of point-dipole sources whichare not symmetrically distributed around the field point. We shall quote theresults, for the uniform case k‖ = 0, of the calculation of the magnetic fieldcomponents Bα at a given lattice point chosen as the origin r‖ = 0, on the layerl of the semi-infinite lattice. If the m− th layer magnetization is Mm, one has

Bα(l,0) = −∑

m,r‖

′ Dαβ(m, r‖)Mβm (13.65)

where the prime on the summation sign means that self-interaction is excluded.This sum was calculated by Kar and Bagchi [14] for the three cubic lattices. Theresults are conveniently displayed in terms of inter-layer coupling coefficients ξm:

Bzl = −

∑

m

ξmMl+m (13.66)


Here Mj is the magnetic moment per basic cell volume on layer j, so that thecoefficients are dimensionless. Let us consider for example α = β = (110) andthe (111) surface of an f.c.c. cubic lattice. The coefficients are displayed inTable 13.1.

Table 13.1: Coefficients for the layer by layer calculation of the dipolar fieldin an f.c.c. lattice with a (111) surface. The magnetization is parallel to thesurface.

ξn n3.9013 00.1434 10.0004 2−2.063 × 10−4 33.0 × 10−9 4

The very fast decrease with m of the field amplitude contributed by parallelplanes m layers away must be attributed to the interference between the infinitepoint dipole fields. The same fast convergence is found for the Fourier compo-nents of the field.

These results can be related to the macroscopic theory. For the slab ge-ometry, the macroscopic local Lorentz field with M parallel to the slab isB = 4πM/3, so that the sum

S = ξ0 + 2

∞∑

n=1

ξn =4π

3(13.67)

which can be verified from Table 13.1.

13.9 Surface magnetism in metals

One can understand the main processes leading to the determination of themagnetic properties of a transition metal surface by appealing again to theHubbard Hamiltonian. We incorporate the changes brought about by the sym-metry breaking due to the surface in a similar way to what we did for insulators.The coordination at the surface is lower, and the boundary conditions on theelectron wave functions of the surface atoms are different from the bulk ones.This implies that the degeneracy and the energy levels of the atomic orbitalswhich determine the Hubbard Hamiltonian parameters change at the surface.If we restrict the model Hamiltonian to a non degenerate basis, we expect to beable to model these changes phenomenologically simply by allowing for differ-ent local parameters at the surface. In the usual tight-binding basis, the choiceof perturbed parameters is often reduced to those involving specifically surfaceatoms. For the one-orbital atomic basis, we assume that the atomic energy level

13.9. SURFACE MAGNETISM IN METALS 325

El and the intra-atomic Coulomb repulsion parameter Ul of the l− th atom atthe surface, as well as the hopping integrals tln which connect surface atomswith each other or with interior atoms, are different in principle from those ofthe bulk. The d-band electrons of a transition metal will be described in whatfollows, within the HF approximation, by the effective one-electron Hamiltonian

H =∑

lσ

εla†lσalσ +

∑

lσ

Ulnlσ〈nl−σ〉 +

∑

l,nσ

tl,na†lσanσ −

∑

l

Ul〈nl↑〉〈nl↓〉 (13.68)

We can now calculate the matrix elements of the resolvent operator, which fora one-electron Hamiltonian, coincides with the Green’s function:

G(ω + iε) = ( ω + iε−H )−1

(13.69)

We already know that we can use the translational symmetry parallel to thesurface, so we go over to the 2d-Fourier-transformed, mixed Bloch-Wannier,basis as we did in the case of insulators and assume also that the hopping hasonly nearest neighbor range. We define the dimensionless matrix g ≡ tG anddivide all energies by t. We call E the ground state energy of a bulk atom andE + ∆ε that corresponding to a surface atom . We also call Ul/t = ul. Theintra-surface hopping integral will be called t‖ and the hopping from the surfaceto the next plane t01. The structure factor of the two-dimensional lattice ofeach plane is Λ(k‖), which for a s.c. lattice and a (010) surface is

Λ(k‖) = 2(coskxa+ cos kza) (13.70)

(the y axis is perpendicular to the planes). Finally we define α‖(k‖) = (t‖/t)Λ(k‖),α = t01/t, ν = (ω −E)/t and ∆E = ∆ε/t. With this notation we find:

g00(ν,k‖) =1

F σ0 (ν,k‖) − ξσ(k‖)α2

(13.71)

whereF σ

0 (ν,k‖) = ν − n0−σus − α‖(k‖) − ∆E (13.72)

The transfer matrix ξσ is the solution of

ξσ = 1/(F σ − ξσ) (13.73)

Here F σ = ν − un−σ − Λ(k‖) only depends on bulk parameters and the bulk

average 〈nbσ〉 = nb

σ . As we already know, for states in the bulk continuumspectrum | ξ |= 1 and it is complex, while for surface states | ξ |< 1. Theeffective one-dimensional perturbation potential at the surface will contain theparameters ∆σ(k‖) = F σ

0 − F σ , α‖(k‖) and α.The layer electron charge and spin,

nl = nl↑ + nl↓

ml = nl↑ − nl↓ (13.74)


must be determined self-consistently. At T = 0 the self-consistency conditionsare

nl = − 1

π

A

4π2

∑

σ

∫

BZ

d2k‖

∫ νF

−∞

Im gσll(ν + iε,k‖)

ml = − 1

π

A

4π2

∑

σ

∫

BZ

d2k‖

∫ νF

−∞

σIm gσll(ν + iε,k‖) (13.75)

where A is the area of the two dimensional lattice cell. For the bulk we knowthe electron concentration and the saturation magnetization, from which we candetermine the set of atomic parameters which inserted in the respective equa-tions (13.75) must yield the known values of n and m. The Fermi energy νF

and the parameters t and u must be consistent with the bulk quantities. Thed-band electron concentration can be estimated from experiments, as well asthe bulk magnetization contribution from the d-bands. One must allow for thecharge and the magnetic polarization of the s, p bands. The parameters u, t andε were obtained in this way by Hasegawa within this model [20]. The parametersfor the surface must fulfill the requirement of local, or quasi-local, neutrality.This condition reduces the number of free surface parameters. In Ref. [21] thevalues of ∆E and us were fixed by the self-consistency HF conditions and therequirement of overall charge neutrality in a region of a few planes near thesurface. The results are not sensitive to the width of the neutrality layer, andone obtains essentially the same parameters upon varying this width from 1 to20 planes. The magnetization ms of the surface plane is the main output of thecalculation.

The effect of the surface on the layer density of states is shown in the curvesobtained by Kalkstein and Soven with the model described here, in a non-self-consistent calculation [22]. Figure 13.8 shows that the local reduction of coor-dination narrows the band at the surface plane, as should be expected. Notethat, in agreement with our previous remarks, the influence of the surface haspractically disappeared already in the third plane. The layer density of states inthe bulk is independent on the layer and has the same form as the bulk densityof states. Note also that the surface density of states is lower than the bulk oneat the band edges, while it peaks at the band center. Finally we remark thatthe density of states at the third layer shows some small oscillations aroundthe bulk curve, which reflect the oscillations of the wave functions according toEq. (13.54). These are the equivalent, for the surface, of the Friedel oscillationsnear a point charge in a metal. The main features described so far are main-tained in more elaborate calculations within the LCAO (Linear Combinationof Atomic Orbitals) with degenerate bands. The concentration of states nearthe band center increases the Stoner parameter UsN(EF ) and favours ferro-magnetism at the surface in metals like Fe,Cr and V with EF near the bandcenter, while the narrowing of the band has the opposite effect in Ni, Pd or Ptwhere EF is near the band edge.

A theory of surface spin waves in metals [23] can be developed with Green’sfunctions methods similar to the ones we presented above for insulators.

13.9. SURFACE MAGNETISM IN METALS 327

Figure 13.8: Layer density of states in tight-binding model of a s.c. transitionmetal. The figure shows the density of states of the (100) surface plane (n=1)and the next two planes.[22]

References

1. Yang, K. Y. , Homa, H. and Schuller, I. K.,(1980) J. Appl. Phys. 63 4066.

2. Celotta, R. J. and Pierce, D. T., (1986) Science 2324 249.

3. Rau, C. and Ecke, G.,(1980) Nuclear Physics Methods in Materials Re-search, edited by Bethge, K. et al., Vieweg, Braunshwewig,

4. Saenz et al., (1987) J. Appl. Phys. 624293.

5. Selzer, S. and Majlis, N. (1983) Phys. Rev. B27, 544.

6. Neel, L.,(1953) Comptes Rendus Acad. Sc. (Paris) 237, 1468.

7. Weller, D., Alvarado, S. F., Gudat, W., Schroder, K. and Campagna, M.(1985) Phys. Rev. Lett. 63, 155.

8. Rau, C. and Jin, C. (1988) J. Appl. Phys. 54, 3667.

9. Binder, K. (1983) “Phase Transitions and CriticalPhenomena” 8, Editedby Domb, C. and Lebowitz, J., Academic Press, London.

10. Diehl, H. W. (1982) J. Appl. Phys. 53, 7914.

11. Damon, R. W. and Eshbach, J. R. (1981) J. Phys. Chem. Solids 19, 308.


12. Selzer, Silvia (1982) Ph.D. Thesis, unpublished, Centro Brasileiro dePesquisas Fısicas, Rio de Janeiro, Brazil.

13. Harriague, S. and Majlis, N. (1971) J. Phys. Soc. (Japan) 31, 1350.

14. Kar, N. and Bagchi, A. (1980) Sol. State. Comm. 33, 645.

15. Wall, H. S. (1967) “Analytic Theory of Continued Fractions”, ChelseaPublishing Co., New York.

16. Selzer, S. and Majlis, N. (1980) J. Mag. and Mag. Mat. 15-18, 1095.

17. Kosterlitz, J. M. and Thouless, D. J. (1973) J. Phys. C 6, 1181.

18. Majlis, N. (1990) “Surface Science”, Proceedings of the 6th. Latin Amer-ican Symposium on Surface Physics, Springer Proceedings in Physics 62,Springer–Verlag, p.443.

19. Dauth, B., Durr W. and Alvarado, S. F. (1987) Surface Science 189/190,729.

20. Hasegawa, H (1980) J. Phys. Soc. Japan 49, 963.

21. Rodrigues, A. M., Majlis, N. and Ure, J. E. (1986) Il Nuovo Cimento 7D,528.

22. Kalkstein, D. and Soven, P. (1981) Surf. Sci. 26, 85.

23. Mathon, J. (1988) Reps. Prog. Phys. 51, 1.

Chapter 14

Two-Magnon Eigenstates

14.1 Introduction

Wortis [1] made a detailed study of the scattering and bound states of twomagnons in a FM, which we shall now survey. As we have mentioned before,this problem had been first solved in 1d by Bethe [2]. The general feature inany dimension is that two spin waves interact through an effective attractivepotential. The question is then whether this potential can sustain bound states.Wortis’ conclusion is that in 1d there are bound states for any value of the totalwave vector of the pair of excitations, in agreement with the results obtainedby Bethe. For two and three dimensions the bound state eigenvalue equationsmust be solved numerically, which makes the complete solution of the problempractically impossible, since there are too many parameters involved. However,it is possible to obtain definite results for some restricted regions of the parame-ter space, and then make some reasonable conjectures on the general behaviourof the solutions. At the end of this chapter we shall review the conclusions ofthis work.

Let us consider the states of the Heisenberg Hamiltonian for a ferromagneticinsulator with two spin-flips located at sites (i, j):

|ψij〉 = S+i S

+j |0〉 (14.1)

where |0〉 is the Weiss ground state, which we shall choose with all spins down.We know already that the one-magnon states with a given wave vector k,

|k〉 =1√N

∑

i

eik·RiS+i |0〉 (14.2)

are exact orthogonal eigenstates with total z component of spin Sztot = −NS+1

and energy

ω(k) = 2SJ(γ(0) − γ(k) ) + γB +E0

329

330 CHAPTER 14. TWO-MAGNON EIGENSTATES

in an external field, where E0 is the ground state energy and γ(k) is the structurefactor. Consider now the family of two plane-wave states

|kk′〉 = A∑

i,j

ei(k·Ri+k′·Rj)|ij〉 (14.3)

For simplicity we shall not use boldface notation for vectors in this chapter whennot necessary for clarity. One can verify that |k0〉 and |0k′〉 are exact eigenstatesof H.

Exercise 14.1Prove that |k0〉 and |0k′〉 are exact eigenstates of H with eigenvalues

E(0, k) = E(k, 0) = E0 + ω(0) + ω(k) (14.4)

This implies that in this particular case we can superpose the two excitationsas if they were mutually independent. However if both k, k′ 6= 0 we have amore interesting situation.

In the first place the wave functions (14.3) are not orthogonal. To see this,consider first the scalar product of the localized two-spin-flip states:


〈12|34〉 = 4S2[δ13δ24 + δ14δ23](1 − δ122S

) (14.5)

In particular we can obtain the normalization constant:

〈12|12〉 = 4S2(1 + δ12)(1 − δ122S

) (14.6)

and then the normalized state can be written as

˜|12〉 =S+(1)S+(2)|0〉

(2S)[(1 + δ12)h2(12)]1/2(14.7)

whereh2(12) ≡ (1 − δ12/2S) (14.8)

The expresssion (14.7) is to be compared with the one-flip state

˜|1〉 =1√2SS+(1)|0〉 (14.9)

With the help of Eq. (14.6) we can now obtain the scalar product we are lookingfor:

14.2. GREEN’S FUNCTION FORMALISM 331


〈k k′|k′′ k′′′〉 = (δkk′′δk′k′′′ + δkk′′′δk′k′′) − 1

NS(δk+k′ ,k′′+k′′′ ) (14.10)

For the case k = k′ = k′′ = k′′′ we have

〈k k|k k〉 = 8S2 − 4S

N(14.11)

We see from Eq. 14.10 that different states with the same total wave vectorhave an overlap ≈ 1

N . This is what Dyson [3] called kinematical interaction.If we impose, as usual, periodic boundary conditions, the number of linearlyindependent states k k′ is the same as that of the states i, j. If i 6= j we haveN(N − 1)/2 different states. If i = j and S > 1/2 we add N more, so the totalis N(N + 1)/2 for S > 1/2 or N(N − 1)/2 for S = 1/2. Observe that since wecan choose N(N + 1)/2 pairs for (k, k′) because there is no special restrictionfor k = k′, then this set is overcomplete in the space of two-spin-flips, at leastfor S = 1/2, so that in this case the set cannot be orthogonalized: N statesmust be linear combinations of the N(N − 3)/2 remaining ones.Let us see what is the effect of H on these states. If neither k nor k′ is zero,then |k k′〉 is not an eigenstate. The Hamiltonian will connect states |k1 k2〉with |k3 k4〉 with the only restriction that k1 +k2 = k3 +k4, which is due to theassumed translation invariance of the infinite lattice. These are the dynamicalinteractions according to Dyson. Their effect is the scattering of the spin-flipexcitations off one another. These processes eventually can lead to bound statesof two spin flips. Let us now study this problem within the Green’s functionformalism [1].

14.2 Green’s function formalism

Since we want to study the eigenstates of the Hamiltonian in the susbpace oftwo-spin flips, we shall calculate expectation values in the ground state (Weissstate for a FM ). We follow Wortis and define the one-spin flip retarded Green’sfunction in a slightly different way from the one we were using so far, as

G1(1; 2; t) = −i〈0|S−(1; t)S+(2; 0)|0〉θ(t) (14.12)

whose Fourier transform satisfies the equation

[ ω − γB + 2S(J(k) − J(0)) ]G1(k, ω) = 2S (14.13)

In this chapter, angular brackets indicate the expectation value in the groundstate.

In a hypercubic lattice of d dimensions with constant a, and with only firstn.n. exchange we have

J(k) = 2

d∑

i=1

cos kia (14.14)


and the magnon frequency is

E(k) = 4Sd∑

i=1

(1 − cos kia) (14.15)

The two-site Green’s function, which will be called G2, is defined as

G2(12; 1′2′; t) = (−i)2θ(t)×〈0 | S−(1; t)S−(2; t)S+(1′; 0)S+(2′; 0) | 0〉 (14.16)

When use is made of the scalar product (14.6) and the definition of h2(12) inEq. (14.8), we obtain the equation of motion

[

i∂

∂t− 2(γB + 2SdJ)

]

G2(12; 1′2′; t)

+∑

3

2SJ(13)G2(32; 1′2′) + 2S∑

3

J(23)G2(13; 1′2′)

+J(12)G2(12; 1′2′) − δ12∑

3

J(13)G2(32; 1′2′)

= (−i)(2S)2[δ11′δ22′ + δ12′δ21′ ]h2(1′2′) (14.17)

The last two terms on the l. h. s. of Eq. (14.17) involve explicitly the coordinates1, 2 and are therefore responsible for the interaction of these two sites. Thefirst terms are proportional to the probability amplitude for the independentpropagation of flips which were present at t = 0 on points 1′, 2′ to the points 1, 2,at a later time t. Then it is natural to introduce the symmetric non-interactingpropagator

Γ2(12; 1′2′) = G1(1, 1′)G1(2, 2

′) +G1(1, 2′)G1(2, 1

′) (14.18)

Exercise 14.4Verify that Γ2 satisfies Eq. (14.17) without the last two terms of the l. h. s.,and with h(12) substituted by 1 on the r. h. s.

Let us write Eq. (14.17) more concisely as

(Ω0 + δΩ)G2 = −4iS2∆h2 (14.19)

where

δΩ(12t; 34t′) = 2[ J(12)δ13δ24 − J(23)δ12δ24 ]δ(t− t′)

∆(12, t; 34) = [ δ13δ24 + δ14δ23 ]δ(t− t′) (14.20)

and Ω0 contains the rest of the integro-differential operator acting on G2 on thel. h. s. of Eq. (14.17):

(Ω0)(12t; 34t′) =

δ(t− t′)[ i∂

∂t+ δ13δ24(−2γB + 2SdJ)

+ 2SJ(13)δ24 + 2SJ(24)δ13] (14.21)


Then one findsΩ0Γ2 = −4S2i∆ (14.22)

Notice that the matrix products above involve two lattice sums and one inte-gration over a time variable, so for two arbitrary matrices A,B:

(AB)(12t; 1′2′, t′) =∫

ds∑

34

A(12, t; 34, s)B(34, s; 1′2′, t′) (14.23)


∆2 = 2∆ (14.24)

From Eq. (14.22) we obtain

Ω0 = −4S2i∆Γ2−1 (14.25)

so that−4S2i∆Γ2

−1G2 = −4S2i∆h2 − δΩG2 (14.26)

Now we multiply both sides of the previous equation by ∆Γ2 from the left anduse Eq. (14.24), obtaining

G2 = Γ2h2 − i

8S2Γ2δΩG2 (14.27)

Let us now introduce an interaction kernel by defining:

Γ2δΩ ≡ K2J (14.28)

where

K2(12t; 34s) =

Γ2(12t; 34s)− 1

2[Γ2(12t; 33s) + Γ2(12t; 44s)] (14.29)

Finally we get

G2 = Γ2h2 − i

8S2K2JG2 (14.30)

In order to arrive at (14.30) we have used the symmetry

G2(abt; ·) = G2(bat; ·) .

We shall now Fourier analyze all the functions involved. Let us start with thefree-magnon-pair propagator Γ2 defined in Eq. (14.18):

Γ2(120; 1′2′t) =1

N2

∑

k

∫

dω

2πG1(k, ω)e−iωteik(1−1′)×

∑

k′

∫

dω′

2πG1(k

′, ω′)e−iω′t ×

eik′(2−2′)eik(1−1′) + eik(1−2′)eik(2−1′) (14.31)


The frequency transform of Γ2 is

Γ2(12; 1′2′; Ω) =

∫

Γ2(120; 1′2′t)dt eiΩt (14.32)

If we substitute both factorsG1 in (14.31) by their expression given in Eq. (14.13),we can perform the time integration, obtaining the factor 2πδ(Ω − ω − ω′), sothat we are left with only one integration over ω in (14.31). The ω integral is

∫

dω1

(ω + iε−Ek)(Ω − ω + iε−Ek′ )(14.33)

The integrand is analytic in the upper half plane, where we can close the contourand use Cauchy formula to obtain

Γ2(12; 1′2′; Ω) =8S2

N2

∑

k

∑

k′

1

Ω − (Ek +Ek′) + iε×

eik′(2−2′)eik(1−1′) + eik(1−2′)eik(2−1′) (14.34)

To simplify we introduce the center of mass (c.o.m.) and relative coordinatesof any pair of points 1 and 2, and correspondingly, the c.o.m. and relative wavevectors, defined respectively as:

R = (1 + 2)/2 , r = 1− 2

K = k1 + k2 , k = (k1 − k2)/2 (14.35)

Due to the symmetry under permutations 1 ↔ 2 , 1′ ↔ 2′, G2 is even in botharguments r, r′. Since K is the sum of two vectors inside the first BZ, it belongsto the extended zone, and in the hypercube this implies −2π/a ≤ Ki < 2π/a.Call the extended zone BZ. We can now write Γ2 as a function of the relativeand c.o.m. coordinates:

Γ2(r1r2t; r′1r

′2t

′) =∑

K∈BZ

eiK(R−R′)

∫

dΩ

2πe−iΩ(t−t′)iΓ2(r, r

′,K,Ω) (14.36)

We analyze K2 in a similar way:

K2(r1r2t; r′1r

′2t

′) =∑

K∈BZ

eiK(R−R′)

∫

dΩ

2πe−iΩ(t−t′)iK2(r, r

′,K,Ω) (14.37)

A phase factor i was introduced into Eqs. 14.36 and 14.37 for convenience.


Γ2(r, r′,K, ω) =

− 8S2N∑

k

cos (k · r) cos (k · r′)ω − 2γB − S(k,K) + iε

(14.38)


and

K2(r, r′,K, ω) = −8S2N×

∑

k

cos (k · r) [ cos (k · r′) − cos (1/2K · r′)]ω − 2γB − S(k,K) + iε

(14.39)

where for a hypercubic d-dimensional lattice

S(k,K) = 8SJd∑

i=1

( 1 − cos (Kia/2) cos (kia) ) (14.40)

and k is the relative wave vector defined before.

Substituting (14.38) and (14.39) into the integral equation (14.27) one gets:

G2(r, r′,K, ω) = Γ2(r, r

′,K, ω)

+J

2S2

∑

δ

K2(r, δ;K,ω)G2(δ, , r′,K, ω) (14.41)

where δ = star of the first n.n. of a site in the lattice.The real poles of G2 are the excitation energies of the ferromagnet, in the

two-spin-flip subspace. We can discard poles of the free propagator Γ2 or thoseof the kernel K2. The last ones, according to Eq. (14.27), are also those of Γ2.In turn, poles of Γ2 are those of the single spin wave propagator G1. The reasonto discard these one-magnon poles is that one can prove that the residues ofG2 at the poles of G1 vanish, so that the only remaining poles of G2 are theexcitation energies of a bound pair of spin waves arising mathematically fromthe zeroes of the secular determinant of the linear system (14.41) as a functionof ω, for fixed total wave vector K. Since by assumption J(δ) has short rangeit suffices to consider one star of nearest neighbour sites. Since G2(r, ·), G2(·, r′)are even, we only consider those vectors in δ with δi > 0. We obtain thelinear system (l, n,m = 1, · · · , d):

d∑

m=1

[

δl,m − 2J

S2K2(l,m) i

]

G2(m,n) = Γ2(l, n)h2(n) (14.42)

If we substitute K2 from Eq. (14.39) we find

det [ 2Sδl,m −B2(l,m) ] = 0 (14.43)

with

B2(l,m) =1

πd

∫ π

0

dk1 · · ·∫ π

0

dkdcos ki cos kj

∆(14.44)

where

∆ = t− iε−d∑

l=1

αl cos kl


t = d− ω − 2γB

8SJαl = cos (Kl/2) (14.45)

We have simplified the notation: ka → k, Ka → K. We first remark that B2

has a cut on the real axis in the interval

S(0,K) ≤ ω − 2γB ≤ S(π,K) (14.46)

This interval contains all the zeroes of ∆, that is all excitation energies

ω(k,K) = 2γB + S(k,K) (14.47)

of scattering eigenstates which result from the superposition of two magnons.In fact the exact energies of the scattering eigenstates differ from the sum ofthe energies of the individual magnons by an amount of order ≈ 1

N , where N =number of spins in the system, so that both excitation spectra coincide in thethermodynamic limit. G2 has the same cut as B2, and both are complex on thecorresponding interval of t. For instance,

Im(B2(i, j)) =

1

πd−1

∫ π

0

dk1 · · ·∫ π

0

dkd cos ki cos kj δ(∆) (14.48)

This cut is the support of the density of two-magnon scattering states. Anybound state must lie therefore in one of the intervals

−∞ < t < −d∑

l=1

αl

d∑

l=1

αl < t <∞ (14.49)

One can define a set of integrals in terms of which B2(i, j) can be expressed.They are

D0(t) =1

πd

∫ π

0

· · ·∫ π

0

d(d)k

∆

Di(t) =1

πd

∫ π

0

· · ·∫ π

0

d(d)k cos ki

∆

Di,j(t) = Dji(t) =

1

πd

∫ π

0

· · ·∫ π

0

d(d)k cos ki cos kj

∆(14.50)

We findB2(i, j) = Di,j(t) −Di(t)αj (14.51)

14.3. ONE DIMENSION 337

The D integrals satisfy the recurrence relations

tD0(t) −d∑

l=1

αlDl(t) = 1

Di(t) −d∑

l=1

αlDi,l(t) = 0 (14.52)

When t is outside the continuum interval, we can expand B2 in inverse powersof t, obtaining:

B2(i, i) =1

2t− α2

i

2t2+

1

t3

(

α2i

8+

1

4

d∑

l=1

α2l

)

+ O(α

t

)4

B2(i, j) = αiαj

(

1

2t3− 1

2t2

)

+ O(α

t

)4

(14.53)

For small αi there is a d-times degenerate solution of the secular equation 14.40:

tB =1

4S(14.54)

14.3 One dimension

In one dimension, the upper and lower limits of the continuum are

t± = ±α ≡ ± cos (K/2)

The D integrals are easily evaluated. One obtains:

D0 =sign(t− α)

(t2 − α2)1/2(14.55)

With the recurrent relations (14.52) one obtains the bound state eigenvalueequation:

2S =t− α2

α2

[ ±t(t2 − α2)1/2

− 1

]

(14.56)

where0 ≤ α ≤ 1 , |t| ≥ α .


To t < −α corresponds the lower sign in Eq. (14.56), so that in this case thereis no solution. In conclusion, there is only one solution for each α for t > α > 0.


Exercise 14.8Show that for S = 1/2 the solution is the singlet

tB =1

2(1 + α2) (14.57)

Then the corresponding excitation energy, referred to the Weiss ground stateenergy and in units of 8SJ , is

ωB/(8SJ) = 1/2(1− α) .

The upper (u) and lower (l) limits of the scattering continuum for two spinwaves in 1d are

ωu, lc

8SJ= 1 ± α (14.58)

In Fig. 14.1 we show the region of the (ω, α) plane which contains the continuum

Figure 14.1: Scattering and bound states in a 1d ferromagnet as a function ofthe total wave vector K of the pair of magnons (α = cosK/2) .

and the dispersion relation of the bound state. The bound state energy lies belowthe continuum, so that clearly the superposition hypothesis, which is the basisof Bloch’s [4] theory of spin waves, is not applicable to the 1d FM.

14.4. TWO DIMENSIONS 339

We define the binding energy of the bound pair as the difference for fixed K,which is equivalent to fixed α, between the bound state energy and the lowerlimit of the continuum, which is

∆B

8SJ= −[

1

2(1 + α2) − α ] ≤ 0 , 0 < α < 1 (14.59)

In one dimension the bound state energy depends only on one parameter, thec.o.m. wave vector. The wave function of the bound state for a given K hasa form which depends on the relative wave vector k, which in the 1d case iscompletely determined for each eigenstate. Since the pair of spin waves is boundthe wave function must be exponentially decreasing as a function of r for larger. This implies that the relative wave vector k must have an imaginary part.One elegant way to obtain the complex k is to consider the bound state energyas the analytic continuation of S(k,K) to complex k. In 1d we have

S(k, α) = 8SJ(1− α cos k) .

We can determine for which k this expression has the same value as the boundstate energy for the same α :

S(k, α) = ωB(α) = 4SJ(1 − α2) (14.60)

and we find

cos k =3 − α2

2α(14.61)

Since |α| ≤ 1 , cos k ≥ 1. Then k = iq for some real q. It is convenient to definean inverse length q = aλ−1 so that

cosh(a

λ

)

=3 − α2

2α(14.62)

and we interpret λ as the distance between both spin-flips for which the wavefunction ΨB(r) of the bound pair is appreciable: for r > λ ,ΨB(r) is expo-nentially small. We see that for α → 0 (K → ±π) λ → 0: in this limitthe support of ΨB(r) reduces to a point. In fact, since the amplitude at r = 0must vanish for S = 1/2, in this case ΨB(r) ≡ 0 : there is no bound state forK = ±π if S = 1/2. However, if α > 0 , ΨB(r) has a finite extension in space,which increases as α → 1, and diverges at α = 1, which again correspods tozero amplitude of ΨB(r) if the length of the chain → ∞.

14.4 Two dimensions

For d = 2 the D integrals can be conveniently expressed in terms of the variableξ defined as:

ξ2 =4α1α2

t2 − (α1 − α2)2, 0 ≤ ξ ≤ 1 (14.63)


The results of the integrals are:

D0(t) =ξK(ξ)

π(α1α2)1/2(14.64)

Di(t) =ξ

παi(α1α2)1/2[ (t+ αj)K(ξ)

−(t+ α1 + α2)Π(β2i , ξ)

]

(14.65)

D12(t) =1

ξπ(α1α2)1/2

[

(2 − ξ2)K(ξ) − 2E(ξ)]

(14.66)

where

β2i =

−2αi

t− αi + αj≤ 0 , β2

1β22 = ξ2 (14.67)

and i, j = 1, 2, i 6= j. The functions K,E,Π are respectively complete ellipticintegrals of the first, second and third kind [5].

In the special case α1 = α2, we have β2i = −ξ. In this case the bound state

equation yields a unique solution for α = 0, with tB > 0 which for J > 0 impliesthat ωB is below the continuum. The analysis of the results for general α1 6= α2

is complicated. Wortis discusses the special case α1 = α2 = α. In Fig. 14.2 weshow a sketch of the continuum region and the bound state dispersion relationsin the ( t, α) plane.

14.5 Summary of results

• In one dimension there is always one bound state. At K = 0 the bindingenergy vanishes but it increases monotonically with | K |. As a conse-quence, the state with two free magnons is unstable against the creationof a bound state, and the superposition hypothesis of free magnons fails.

• In two dimensions, there is one bound state at K = 0 below the lowerlimit of the continuum, so that the conclusion is the same as in 1d. Forα1 = α2 there is a second bound state which requires a threshold of K.

• In three dimensions there are no bound states near K = 0 but there canbe up to 3 bound states for K 6= 0, although numerical estimates so farindicate that this happens at relatively large values of K. One concludesthat in 3d magnons behave as free excitations if the one-magnon-statewave vectors are not near the BZ boundary. Since this is the case atnot too high temperatures, the superposition hypothesis of Bloch, andaccordingly spin wave theory, are also valid at low T .

14.6. ANISOTROPY EFFECTS 341

Figure 14.2: Two magnon states in a 2d FM, for α1 = α2 = α . The two boundstates are degenerate at α = 0. The lower branch merges with the continuum atα = 1

2S [ 4π − 1] . For t < 0 , which corresponds to states above the continuum,

there are no bound states.

14.6 Anisotropy effects

We consider now materials with highly anisotropic exchange interactions, de-scribed by

H = −γB∑

1

Sz(1)

−∑

12

J(12)[Sz(1)Sz(2) + ∆S+(1)S−(2)] (14.68)

We see that the limit ∆ = 0 is the Ising Hamiltonian while the Heisenbergisotropic case is obtained with ∆ = 1. For all intermediate values in that in-


terval the one-magnon spectrum has a gap, which would make LRO possibleat finite temperatures even in 1d, were it not for the kink instability alreadydiscussed in Chap. 12. For the Ising model it is easy to show that two separatespin flips interact with an attractive effective force:

Exercise 14.9Show that for an Ising hypercubic lattice with first n.n. interactions the excita-tion energy for two spin flips separated by more than a lattice constant is

Econt = 2γB + 2νJS (14.69)

while if they lie on adjacent sites the energy is

Ebound = Econt − 2JS (14.70)

which has degeneracy d.

Since the binding energy is d-independent, we expect these excitations to playa role in particular in 3d sytems with high anisotropy.For 0 < ∆ < 1 one can basically repeat all the previous algebra. We can define

K2(12t; 34s) =

Γ2(12t; 34S)− ∆

2[ Γ2(12t; 33s) + Γ2(12t; 44s) ] (14.71)

where the one-magnon propagator factors in Γ2 are those of the anisotropicsystem. The only change in the previous formulae is the substitution

αi → ∆ cos

(

Ki

2

)

0 ≤ αi < ∆ ≤ 1 (14.72)

Observe that in 1d the bound state is always below the lower limit of the contin-uum, which now has a finite gap proportional to η = 1−∆. Then the populationof bound states is exponentially higher at low non-zero T than that of the con-tinuum states.

Exercise 14.10Show that in 1d

Ebound −E0 = 2γB + 2SJ

(

1 − ∆2 cos2K

2

)

(14.73)

which is the result obtained by Orbach [6] in his extension of the Bethe-Ansatztheory to include anisotropic exchange.

14.6. ANISOTROPY EFFECTS 343

In 3d Wortis [1] finds that for

0 ≤ ∆ <0.5163

2S + 0.5163(14.74)

there will be a bound state for K = 0 with energy lower than the lower contin-uum limit (with anisotropy) so that bound states dominate the thermodynamicsat low T in a 3d highly anisotropic FM.

References

1. Wortis, M. (1963) Phys. Rev. 132, 85.

2. Bethe, H. A. (1931) Z. Physik 71, 205.

3. Dyson, F.J. (1956) Phys. Rev. 102, 1217.

4. Bloch, F. (1931) Z. Physik 71, 205.

5. Abramowitz M. and I. Stegun (1965) Handbook of Mathematical Func-tions, Dover Publications, Inc., New York, U.S.A.

6. Orbach, R. (1958) Phys. Rev. 112, 309.

Chapter 15

Other Interactions

15.1 Introduction

In Chapter 4 we described the Holstein-Primakoff expansion of the spin oper-ators in a series of products of boson operators, and we explicitly wrote thatexpansion up to fourth order terms. Clearly terms of order higher that the sec-ond give rise to magnon-magnon scattering and accordingly to a finite magnonlifetime. In the classic paper by Dyson [1] the low temperature scattering crosssection is calculated to various orders. We remark that from the expansion ofthe factors f (Eq. (4.17)) in the exchange Hamiltonian in Chap. 4 one can onlyget terms which contain the same number of creation and annihilation opera-tors, so that each term conserves the total Sz. The HP expansion of the dipolarHamiltonian, instead, can generate terms with either an even or odd number ofoperators. Clearly the odd terms do not conserve Sz.

If we classify the interactions which are responsible for the finite lifetime ofmagnons according to the number of magnon operators they involve, we have inthe first place interactions of magnons with static impurities or structural de-fects in general. In particular surface pits are the basic source of the low ordertwo-magnon processes in which the total pseudo-linear-momentum of the spinsystem is not conserved, the excess being transferred to the linear momentumof the lattice. An interaction Hamiltonian with impurities of the form

Himp =∑

k,q

Fq b†k+qbk + c.c. (15.1)

was studied by Sparks et al. [2]. We see that (15.1) conserves the z componentof the total spin Sz of the system. In the case that one of the magnons involvedis a uniform precession mode, this interaction does not conserve S2, instead.

The magnon-phonon interaction is also quadratic in the magnon operators.In dielectrics it can arise either from exchange or single-site anisotropy terms inthe Hamiltonian, since the parameters of these interactions can be modulated bythe ionic displacements. In the rigid-ion approximation, one assumes that the

345

346 CHAPTER 15. OTHER INTERACTIONS

electrons follow the nuclear displacements but the ions stay in the ground state,although in fact one could consider as well the possibility that they make virtualtransitions to excited states, a correction to the magnon-phonon matrix elementwhich has been considered by Sinha and Upadhayaya [3]. In metals, theelectron-phonon interaction gives rise to an effective magnon-phonon couplingwhich was obtained in the RPA [4, 5]

The next order is provided by the dipolar interaction. From the expansion

of the terms in H(2)dip in Eq. (6.32) we obtain cubic terms. These generate the

processes of confluence and splitting of magnons. In a confluence, two magnonsare destroyed and another one is created with conservation of the total wavevector. The time-reversed process is the splitting of one magnon into a pair.These processes do not conserve either Sz or, in general, S2.

In fourth order we have processes of scattering of two magnons .All the above mentioned interactions are eventually responsible for the

relaxation of the perturbed magnetization (ferromagnetic relaxation ) to itsequilibrium value, that is, they provide equilibration processes. One convenientway to model relaxation phenomena in the long wave limit is the use of amacroscopic differential equation for the magnetization based on Bloch’s torqueequation which was introduced in Chap. 4.:

dM(r, t)

dt= γM(r, t) ∧ Beff (r, t) (15.2)

supplemented with phenomenological damping terms adequate to the variousprocesses involved.

Several forms of the damping terms have been proposed. Landau and Lifshitz[6] in 1935 chose the form

− 1

2T

1

| M |2 M(r, t) ∧ [ M(r, t) ∧ Beff (r, t) ] (15.3)

The factor 2 in the time constant is included because in general T is reservedfor the time constant of the energy, which is proportional to the square of themagnetization. This rate of change is perpendicular to M, so that dM2 = 0.Therefore, the tip of M spirals on the surface | M |= constant. Let us write

M = M‖ + m (15.4)

separating the longitudinal component, parallel to the external field B, and theassumed small- transverse part and rewrite (15.3) as

− 1

2T

M‖Beff

| M |2 m +m2

2TM2Beff (15.5)

Then to first order in m/M the motion of M is a spiralling approach to thedirection of Beff during which M is constant, while the transverse componentdecreases and the longitudinal one increases.

Another form initially proposed by Felix Bloch to describe nuclear magnetic

15.2. TWO-MAGNON INTERACTION 347

relaxation was later adapted by N. Bloembergen to describe ferromagnetic re-laxation [7]:

dMz

dt= −γ(M ∧ Beff )z −

Mz −M

2T1

dMx,y

dt= −γ(M ∧ Beff )x,y − Mx,y

2T2(15.6)

Let us consider a few examples of cases where one or the other of these formsis adequate to describe the relaxation processes.

In a two-magnon process in which a uniform precession magnon is destroyedand another with a finite k is created, M and m change while Mz is kept con-stant. This contributes to the Bloch-Bloembergen (BB) time T2.

Consider now a 3-magnon process in which a k = 0 and a k 6= 0 magnonsare destroyed, while another is created with k 6= 0. Then the total numberof uniform magnons n0 decreases and the total number of magnons changes, sothat Mz changes (see Chap. 7). At the same time, the total number of magnonswith k 6= 0 does not change, so that | M | remains constant. This is correctlydescribed by the Landau-Lifshitz (LL) term.

Let us consider now a 3 magnon confluence process, in which two magnonswith respective wave vectors k1, k2 are annihilated and one with wave vectork1 +k2 is created. If none of these wave vectors is zero, n0 does not change, andaccordingly (see Chap. 6) m2 remains the same, while M2 and Mz change, sothat we can describe this process with Bloembergen’s T2 but not with the LLform.

Exercise 15.1Analyze the following examples and establish which of the above forms of damp-ing term applies:

1. In a 3 magnon process, one magnon with k = 0 and one with k 6= 0 areannihilated, while another one with k 6= 0 is created.

2. In a two-magnon-one-phonon process one uniform precession magnon isannihilated, one k 6= 0 magnon is created and one phonon is either createdor annihilated.

We shall now consider the different interactions separately.

15.2 Two-magnon interaction

In this section we want to describe briefly the kind of processes which are mainlyresponsible for the relaxation of the uniform mode. An interaction Hamiltonianbased on the assumption that the main cause is the scattering with imperfectionsat the surface was given in Eq. (15.1). The matrix elements Fq were obtainedby Sparks et al. [8].


Let us consider the possible channels for relaxation of the uniform mode.In the experiments a ferromagnetic sample is placed in an external magneticfield B large enough to saturate the magnetization along the z axis. The totalmagnetic moment will precess around the field at the Larmoor angular frequencyωc = γB. If B is typically a few thousand oersteds ωc ≈ 1011 sec−1, or afrequency ν = ωc/(2π) ≈ 10 kMc, which lies within the X band. A transverser. f. field h of the same frequency can transfer energy to the magnet, increasingits potential energy W = −VM ·B in the static field, thereby increasing theinclination of the magnetization in relation to the z axis. If at some instant ther. f. field is turned off, the system is left in a state out of equilibrium. Theinteractions we are considering are responsible for the subsequent relaxationto equilibrium, which implies the recovery of the initial precessional motion.Clearly the interactions which decrease the transverse component m and/orincrease the longitudinal component Mz of the magnetization will contribute tothe equilibration process. Experimentally, transitions of the first kind, whichconserveMz, are found to be faster, with a characeristic time constant which it iscustomary to call T2. In a subsequent, slower stage, Mz increases, with anothertime constant called T1. In Fig. 15.1 we show schematically the spiralling motionof the vector M in the relaxation stage with Mz = constant and the subsequenthelical motion to recover the saturation value Mz = Ms a process which iscompatible with the BB damping term.

Fig. 15.2 shows a different path to equilibrium, in which | M |= constant,as in LL damping. We notice that the reverse path to that in Fig. 15.2 wouldbe followed by M as the r. f. field is turned on, since a uniform field can onlyexcite uniform precession magnons, and they conserve M2 (see Chap. 7). Mz

Figure 15.1: Relaxation of M. From point B to C Mz is approximately con-served (T2 BB stage). After instant C the basic process is the recovery of thesaturated Mz, as in the T1 BB stage.

decreases because the spatially uniform r. f. field excites k = 0 magnons. Eachnew magnon decreases VMz by one unit of h. Immediately after the r. f. field

15.3. THREE-MAGNON PROCESSES 349

Figure 15.2: Relaxation of M with conservation of M2 as in the LL damping .At B the r. f. field is switched off. At A the equilibrium saturated Ms has beenrecovered.

is switched off the magnet begins to relax to the equilibrium uniform precessionfollowing in different scales of time one among several alternative competingchannels. One such is the two-magnon process we described before. Aftera time comparable with T2 other processes begin to dominate the relaxation.Experiments show that the fast relaxation is effectively dominated by the surfacepit scattering. In effect, the resonance linewidth increases with the radius of thesurface pits, which can be controlled by the polishing treatment of the samplesurface, in agreement with the theory [2] based on this mechanism. A detaileddiscussion of the calculation of the linewidth based on this model can be foundin Sparks’ book [8].

15.3 Three-magnon processes

The dipolar interaction is the only source of terms with three magnon operators.

From H(1)d in Eq. (6.31) we get a cubic term

Hcubic =3γ2

√2S

2

∑

l6=m

Flm[ a†l alal − ala†mam ] + h. c. (15.7)


We remind that the expresion of F (R) is, from Eq. (6.14),

F (R) =(X − iY )Z

R5(15.8)

We assume our system has the reflection symmetry z → −z, so that the first sumin Eq. (15.7) vanishes, unless the point considered be very near to the surface(at a distance of only a few lattice planes from it). With this simplificationwe are left with only the second sum in (15.7), which in terms of spin waveoperators reads:

Hcubic = −3γ2√

2S

2

∑

q,k

[

Fqb†k+qbqbk + F ∗

q b†kb

†qbk+q

]

(15.9)

This describes processes of confluence of two magnons into one or splitting ofone into two, where q is the pseudo-linear-momentum transfer. Let us simplifythe notation, by calling the wave vectors, ki, i = 1, 2, 3:

Hcubic =∑

1,2,3

∆(k1 − k2 − k3)[

C123b†1b2b3 + C∗

123b†3b

†2b1

]

(15.10)

where C is the matrix element of the perturbing Hamiltonian and ∆(k) is theKronecker’s δk,0. We neglect the effects of the Bogoliubov transformation, sowe calculate the matrix elements of Hcubic between states characterized by agiven number of spin-waves of each wave-vector ki. Let us consider for examplethe confluence event represented in Fig. 15.3(a):

〈n1 + 1, n2 − 1, n3 − 1 | Hcubic | n1, n2, n3〉= C∗

123b†1b2b3 + C∗

132b†1b3b2 (15.11)

The transition probability per unit time (TP) from a given initial state i toa final state f due to a perturbation potential V is given in the first Bornapproximation by the Golden Rule [9]

TP =2π

h| 〈i | V | f〉 |2 δ(hωf − hωi) (15.12)

Let us consider now a given k1 and an initial state where the number of magnonsk1 at time t is n1(t) and n2, n3 are the thermal equilibrium values given by Bose-Einstein distribution function. Then the total transition probability to any finalstate is the sum

TP =2π

h2

∑

f

| 〈i | V | f〉 |2 δ(ωf − ωi)(n)∆(K) (15.13)

where we changed from energy to frequency in the argument of δ. The factor∆(K) is the Kronecker ∆ which takes account of wave-vector conservation,while the notation (n) contains all thermodynamic weight factors for the process

15.3. THREE-MAGNON PROCESSES 351

Figure 15.3: Three magnon processes involving states k1,k2 and k3.

involved. Those contributions which increase (decrease) n1 must be added to(subtracted from) the time rate of change of n1(t), so that the contributionsfrom confluence and splitting events involving two given sates can be addedtogether, with their corresponding sign. Therefore we arrive at

dn1

dt=

1

2

∑

2,3

[ (n1 + 1)n2n3 − n1(n2 + 1)(n3 + 1) ] ×

| C312 + C321 |2 ∆(k1 − k2 − k3)δ(ω1 − ω2 − ω3)

+∑

2,3

[ (n1 + 1)(n2 + 1)n3 − n1n2(n3 + 1) ] ×

| C312 + C321 |2 ∆(k3 − k1 − k2)δ(ω3 − ω1 − ω2) (15.14)

where the factor 1/2 in the first sum corrects overcounting. The terms insquare brackets are what we called (n) in Eq. (15.13). One verifies that ifthe three magnon populations are in equilibrium the expression (n) vanishes inboth terms. If wew call (n) the result of substituting in (n) the Bose-Einsteindistributions, we have (n) ≡ 0, so that we can always subtract (n) from (n) inall expressions above. Let us consider processes described by Fig. 15.3 (a) and


(b), which are a confluence of k1 and k2 into k3 and the time reversed splittingprocess.

Exercise 15.2Verify the vanishing of the statistical factor (n) due to energy conservation.

Upon subtracting (n) from the original statistical factor, and using again energyconservation, one finds that

(n)conf = n1+2n2eβhω2

(

1 − eβhω1)

(n1 − n1) (15.15)

Notice that the expression above is independent of k3. One verifies that thecases (c) and (d) of Fig. 15.3 can also be cast in a similar form, which yields

dn1

dt=

(

1

Tconf+

1

Tsplit

)

(n1 − n1) (15.16)

where ( substituting the sum over k2 by an integration over the BZ, as usual)

1

Tconf= V

(

1

2πh

)2(

eβhω1 − 1)

∫

dk2 ×

| Ck1 + Ck2 |2 eβhωk2 δ(ωk1 + ωk2 − ωk1−k2) (15.17)

and

1

Tsplit=

(

1

2πh

)2(

eβhω1 − 1)

×∫

dk2V | Ck2 + Ck1−k2 |2 δ(ωk1 − ωk2 − ωk1−k2) (15.18)


Each of these relaxation times is predominant for given intervals of values of k1

which depend, in turn, on the temperature range. Detailed calculations weremade by Sparks et al. [2] and by Schlomann et al. [10]. It turns out that phasespace restrictions imposed by energy and momentum conservation lead to thevanishing of the transition probability for the confluence process as k1 → 0 [8].We close this section with the calculation of the Fourier transform

Fk =1

N

∑

R

′ eik·R (X − iY )Z

R5(15.19)

As we did in Chapter 7 for similar calculations, we expand the plane wave inspherical harmonics and change the sum into an integral, since we are interestedin the long-wave limit ka 1, a = lattice constant. We shall also assumekR 1, where R is the sample diameter, and we shall neglect surface effects,

15.4. MAGNON-PHONON INTERACTION 353

so that every point is very far from the surface and we can assume translationsymmetry. We recognize that

(X − iY )Z

R2= AY −1

2 (Ω)

where A is a coefficient, and substitute the upper limit in the r integral by ∞,obtaining

Fk ≈ 4πAY −12 (Ωk)

∫ ∞

a

drj2(kr)

r(15.20)

Finally, we get

Ck = −8πµB√N

√

µBMs

a3(15.21)

15.4 Magnon-phonon interaction

Both the effective exchange integral and the dipolar interaction tensor dependon the distance between interacting spins, which oscillates, even at T = 0,due to thermal fluctuations of the positions of ions. The distance Rab betweenneighbouring sites a, b can be written as

Rab = Rb −Ra = R(0)ab + δRb − δRa (15.22)

where δRa,b are the instantaneous displacements of the respective ions. Weexpand these displacements as linear combinations of phonon creation and an-nihilation operators. For simplicity consider that the crystal has only one ionper unit cell. Then the α component of the displacement of ion l is [12]

δRα,l =

√

(

h

NMat

)

∑

s,k

1√

2 ωs(k)

[

e(s)α (k)eik·R0l as(k)

+ e(s)α (k)e−ik·R0l a†s(k)

]

(15.23)

where a†s(k) (as(k)) creates (annihilates) a phonon of wave-vector k inside theBZ, with polarization s and corresponding dispersion relation ωs(k). We havetaken for simplicity a lattice with only one kind of atoms, so all ionic massesMat are equal. The phases of the eigenvectors of the dynamical crystal matrixcan be chosen so that

e(s)∗α (k) = e(s)α (−k) (15.24)

Then if we call

gs(k) = −ie(s)(k)

√

(

h

Mat

)

(15.25)

we can simplify the expansion (15.23):

δRl =1√N

∑

s,k

gs(k)(

a†s(k) − as(−k))

eik·R0l (15.26)


We return to the Heisenberg Hamiltonian and expand the effective exchangeintegral J(Rlm) in a Taylor series around the equilibrium distance R0

lm:

J(Rlm) = J(R0lm) + ∇RJ(Rlm)|R0 · (δRl − δRm) + · · · (15.27)

We substitute (15.27) and the expansion (15.26) in the exchange Hamiltonian,to obtain the magnon-phonon interaction Hamiltonian:

Hmp =∑

k,q

φsk,qb

†k−qbk

(

a†qs − a−qs

)

(15.28)

where

φsk,q =

4S√N

∑

∆

∇J(∆) · gs(k)(

eik·∆ − 1) (

1 − e−iq·∆)

(15.29)

and ∆ is as usual the star of nearest neighbours. With this interaction wecan calculate the lifetime and the energy renormalization of both phonons andmagnons. This is particularly relevant in a non-equilibrium situation such that,due to some external perturbation, there is a deviation of the magnon popu-lation from the thermal equilibrium average, which can decay via the emissionof phonons. Likewise, a deviation of the phonon population can decay throughscattering with magnons. The relaxation of the energy excess accumulated ineither the phonon or magnon system would be controlled by the spin-lattice (ormagnon-phonon) τmp relaxation time if the equilibration times of the separatesubsystems, τmm and τpp , were considerably longer than τmp [13, 14]. Pertur-bation theory calculations of the magnon-phonon lifetime were carried out bySinha and Upadhyaya [3] and by Pytte [11]. In a two-sublattice ferrimagnetor in an AFM the magnon-phonon interaction involves also the optical phononbranches.[17, 18]

Since the coupling Hamiltonian of Eq. (15.28) is linear in the phonon opera-tors the lowest order in which one finds a self energy correction for the magnonis the second. The problem is formally similar to the electron-phonon system[15]. One can use perturbation theory to second order and find the complexcorrection to the unperturbed energy of a magnon. It is also possible to extendthe range of temperatures for this calculation if one maintains the original formof the Heisenberg Hamiltonian in terms of the spin operators and treats theproblem with the Green function or equation of motion formalism, applyingsome decoupling scheme to the -by now familiar- hierarchy of equations (seeChap. 6) for the spin operators, and treats the phonon system in the usualperturbation theory. This program was developed by Pytte [11], who obtainedthe energy renormalization and the lifetime of magnons and phonons both atlow temperatures and near the critical temperature of a FM. In both cases theunperturbed energies of both types of excitations are very different from eachother. There is however some interval of values of temperature and k such thatthe dispersion relations of magnons and phonons intersect, and in this case onecannot apply the usual perturbation theory without first eliminating this de-generacy. This implies the consideration of the mixed magnon-phonon modes

15.5. BILINEAR MAGNON-PHONON INTERACTION 355

(or magneto-elastic modes in the long k limit). We address this problem innext section. We shall see that in this case a simpler Hamiltonian that (15.28)applies.

The local single-ion anisotropy energy generates a one-phonon-two-magnoninteraction [16]. Consider the simplest case of uniaxial anisotropy

Va = −∑

i

D(Ri) (Szi )2 (15.30)

Suppose we have a cubic crystal. Then if the Ri are equilibrium coordinates,this term should be absent, so that

D(R(0)i ) = 0, ∀ i

However if all sites are in motion, there may be distorsions at each site, andsome anisotropy might appear as a result. If site µ is displaced by δRµ, wecan expand the anisotropy parameter in the relative displacements of the firstnearest neighbours of each site as

D =∑

µ,ν

Dµν∆µν , ∆µν = δRνµ − δRν

0 (15.31)

Here index µ runs over the star of first n. n. and ν = x, y, z. We may considerthe case in which a uniaxial distorsion is generated, that is, when only Dzz 6= 0.There may be a small static uniaxial distortion, so that we write in general

∆zz = ∆zzstat + ∆zz

phon (15.32)

Now we expand (Szi )

2in magnon operators up to second order terms. For the

strain component we can take the small k limit

∆zz =∂δRz

∂z(15.33)

and expand δRz as in Eq. (15.23) to obtain a two-magnon-one-phonon interac-tion. This interaction was experimentally observed in the paramagnetic reso-nance of Fe3+ in Ytrium Gallium Garnet [16].

15.5 Bilinear magnon-phonon interaction

Let us return to the dipolar Hamiltonian. From Eq. (6.31) we get a linear termin the spin-wave operators:

−3γ2S√

2S

4

∑

n,m

′F (Rlm) al + h. c. (15.34)


The sum over m vanishes when the sites are in their equibibrium positions,which is why we disregarded the linear term until now. However, if we expandF (R) in Taylor series around the equilibrium position

F (Rlm) = F (R0lm) + ∇RF (Rlm)|R0 · (δRl − δRm) + · · · (15.35)

and expand δR in phonon operators we obtain a Hamiltonian which is bilinearin the magnon and phonon operators:

H(1)mp =

∑

q

b†q(

G−q a†q +G∗

q aq

)

(15.36)

This interaction leads to mixing of both types of excitation involved. Insofar asthe difference in their energies is larger than the coupling constant G one cantreat its effects as a perturbation. This is not correct, however, in a region ofq where both dispersion relations cross. The point in the ω, q plane where thisoccurs, if it were the case, is called the nominal crossing, since in reality thecrossing does not occur, precisely because both modes mix and the dispersionrelations resulting from the exact treatment of Hamiltonian (15.36) avoid eachother in that region.

The acoustic-phonon dispersion relations are linear in the wave vector atsmall k, and along crystalline symmetry axes they can be written as

ω(s)k ≈ csk

In general one has three different phonon energies for an arbitrary k, butalong a symmetry axis two polarizations are degenerate, and for small k corre-spond to shear waves, while the remaining one is a longitudinal mode describingcompression waves. A typical sound velocity is cs ≈ 3 · 105 cm/sec. On theother hand, Eq. (6.58) gives for a spherical sample at long wavelengths, and forθk = π/2 the expression

ω2k ≈ γ2B2 +

γD

hk2

(

2B +4πM

3

)

(15.37)

where B is the external field and the exchange energy is

ωe(k) ≈ Dk2.

If there is no demagnetization field, as in a disk with M on the plane of thedisk, and if k ‖ M, we have a pure exchange dispersion relation if B = 0. Oneverifies in this case that both curves intersect if k ≈ 106cm−1 at a frequencyω ≈ 1012 1/sec. Suppose B is not zero, there is no demagnetization field, andk ‖ B. Then the frequency for nominal crossing can increase. For B = 100 oer-steds, the cross-over occurs at ω ≈ 2 ·109 rad/cm , or a frequency of 300 Mc/sec,at k ≈ 106 cm−1, where the exchange contribution is negligible.

In general, the magnon frequency is finite at k → 0 while the acousticphonons have a dispersion relation linear in k, so that for very small k both

15.5. BILINEAR MAGNON-PHONON INTERACTION 357

excitations remain well defined, and magnons are the upper branch. For in-tervals of k as analyzed above there is a region where they turn into mixedmagneto-elastic waves. For higher k they become again separate excitationsbut they have interchanged roles: the upper branch is now phonon-type, andthe lower magnon-type.One can easily verify this description. Call the non-interacting dispersion rela-tion ωk for magnons and Ωk for phonons. Then we write the non-interactingHamiltonian for both excitations as

H0 =∑

k

ωkb†kbk +

∑

k

Ωka†kak (15.38)

If we add the bilinear magnon-phonon Hamiltonian of Eq. (15.36) to Eq. (15.38)we can easily diagonalize the perturbed Hamiltonian.

Exercise 15.4Add the bilinear magnon-phonon Hamiltonian of Eq. (15.36) to the harmonicmagnon and phonon unperturbed Hamiltonian (15.38) and diagonalize the per-turbed Hamiltonian. Analyze the resultant modes both near and far from thenominal crossing.

References

1. Dyson, F. J. (1956) Phys. Rev. 102, 127.

2. Sparks, M., Loudon, R. and Kittel, C. (1961) Phys. Rev. 122, 791.

3. Sinha, K. P. and Upadhyaya, U. N. (1962) Phys. Rev. 127, 432.

4. Rajagopal, A. K. and Joshi, S. K. (1967) Phys. Lett. 24A, 95.

5. Alascio, Blas and Lopez, Arturo, (1970) J. Phys. Chem. Solids 31, 1647.

6. Landau, L. and Lifshitz, E. (1935) Phys. Sov. Union 8, 153.

7. Bloembergen, N. (1956) Proc. IRE 44, 1259.

8. Sparks, M. (1964) Ferromagnetic Relaxation Theory, Mc Graw-Hill BookCo., New York.

9. Messiah, A. (1960) Mecanique Quantique, Dunod, Paris.

10. Schlomann, E. and Joseph, R. I. (1961) J. Appl. Phys. 32, 165S andidem, 1006.

11. Pytte, E. (1965) Ann. Phys. (N. Y. 32, 377.

12. Callaway, Joseph (1976) “Quantum Theory of the Solid State”, AcademicPress, Inc., New York.


13. Sinha, K. P. and Kumar, N. (1980) “Interactions in Magnetically OrderedSolids”, Oxford University Press.

14. Akhiezer, A. I. (1946) J. Phys. (USSR) 10, 217.

15. Abrikosov, A. A., Gorkov, L. P. and Dzyaloshinski, I. E. “Methods ofQuantum Field Theory in Statistical Physics”, (1975), Dover PublicationsInc., New York.

16. Geschwind, S. (1961) Phys. Rev. 12,363.

17. Kasuya, T. and LeCraw, R. C., (1961) Phys. Rev. Lett. 6, 223.

18. Shukla, G. C. and Sinha, K. P. (1967) Can. J. Phys. 45, 2719.

Appendix A

Group Theory

A.1 Definition of group

Definition A.1 GrupoidA groupoid is a pair consisting of a non-empty set G called the carrier and abinary operator µ in G: if g, h ∈ G then (g, h)µ, the result of the binary operatoron that pair of elements of G, is also in G, and we write simply gh = f ∈ G.The order of G is the number of its elements. We say that the groupoid isassociative if for any threee elements a, b, c ∈ Gwe have

(ab)c = a(bc) (A.1)

Definition A.2 SemigroupAn associative groupoid is called a semigroup

An example is the so called renormalization group in statistical physics.If for any pair (a, b) ∈ G we have

ab = ba (A.2)

then G is commutative.An element e is called an identity element if

∀g ∈ G, eg = ge = g .

Definition A.3 MonoidA semigroup with an identity is called a monoid.

Let us call the identity simply 1. In a monoid, we call an inverse of g an h whichsatisfies

gh = 1 = hg (A.3)

and this implies that g is reciprocally the inverse of h. An element cannot havemore than one inverse.

359

360 APPENDIX A. GROUP THEORY

Example A.1 Example of monoidConsider the set MX of all the mappings of a set X on itself, and the binaryoperation consisting of the composition of mappings. Then if X is a non-emptyset, MX is a monoid.

Definition A.4 A mapping S → T is onto if every t ∈ T has at least onepre-image s ∈ S. If every t has one and only one pre-image we say it is ontoand one-to-one, and the mapping is called matching or bijective.

Definition A.5 GroupA group is a monoid in which every element has an inverse.

Definition A.6 HomomorphismA homomorphism is a mapping θ of a grupoid (G,α) into a grupoid (H, β) suchthat

(

(g1, g2)α

)

θ= (g1θ, g2θ)β , ∀g1, g2 ∈ G (A.4)

where θ is the law of transformation from elements of G to elements of H, andα(β) are respectively the association operations between elements in G(H).

Definition A.7 IsomorphismAn isomorphism is a homomorphism which is both one to one and onto.

Clearly the definition applies also to groups, and in this case the identity of Gcorresponds to the identity of H and inverses of elements are transformed intoinverses of the transformed elements. All mutually isomorphic groups can bethought of as copies or realizations of an abstract group, which is completelycharacterized by the multiplication table between its elements. In particularwe shall deal with mappings of a set of points into itself as a realization of anabstract group.

A.2 Group representations

Let us consider a real vector space Rn of n dimensions and all the possible linearhomogeneous transformations of this space into itself. The set Σ(G) of lineartransformations which is isomorphous to a given abstract group G is a specialrealization thereof which is called a representation. To each s ∈ G we associatea linear transformation U(s) : Rn → Rn of the vector space such that

1. U(s)U(t) = U(st)

2. U(e) = 1

3. U(s−1) = (U(s))−1

Upon choosing a particular basis for the space Rn, all n × n matrices whichrepresent the transformations must have determinant 6= 0 due to condition 3)above. A change of basis of the space R induces a cordinate transformation Cwhereby

U(s) ⇒ CU(s)C−1 .

A.2. GROUP REPRESENTATIONS 361

A.2.1 Reducibility

Let us consider now such a family Σ(G) of transformations Σ : R → R. Asubspace R′ which is a subset of R is invariant under Σ if ∀U ∈ Σ , U(R′) → R′.If the representation is unitary, that is if U † = U−1 ∀U ∈ Σ, the complementsubspace R′′ = R − R′ is also invariant under Σ (but in general this might notbe the case). If both R′ and R′′ are invariant all matrices U ∈ Σ contain twoindependent blocks which have respectively elements only in R′ or R′′.We saythat Σ reduces to the sum of Σ′ and Σ′′. The original representation is calledreducible. If one chooses bases for R′ and R′′ which are mutually orthogonaleach matrix of Σ deblocks simultaneously into two blocks of dimensions n′, n′′,where n = n′ + n′′ and n′ is the dimension of R′, n′′ that of R′′. For complexvector spaces, these concepts can be applied to the case of unitary vector spaces.

Definition A.8 A unitary vector space is one in which a scalar product of twovectors r, v exists, defined in general as

(r, v) = riAikvk (A.5)

with A† = A, which ensures that the norm of a vector is real: (r, r)=real. If forany vector r 6= 0 the norm is positive, the eigenvalues of A are all positive. Weshall consider this to be the case.

Theorem A.1 If R is a unitary vector space invariant under a group of trans-formations then either R is irreducible under G or it can be completely reducedto a sum of mutually orthogonal subspaces which are also invariant under G.This property is invariant under unitary transformations of the basis of R.

In a representation D of a group G each s ∈ G is mapped into a matrix D(s).

Definition A.9 Characters of a representationWe call

χ(s) = TrD(s) (A.6)

the character of s in the representation D.

Since the trace is invariant under unitary transformations, so are the characters.If the representation is reducible such that

D =r∑

i=1

D(i) (A.7)

then the characters satisfy

χD(s) =

r∑

i=1

χD(i) (s) (A.8)

Definition A.10 Irreducible representationsIf a representation D cannot be de-blocked into smaller dimensional matrices,it is called irreducible.


A.3 Orthogonality relations

Consider two different representations Dλ, Dµ of a group G of transformationsof a unitary space R and let h be the order (assumed finite) of G. Then wetake for every transformation S ∈ G and for fixed pairs of indices (kl), (ij)the product Dλ(S−1)klD

µ(S)ij and sum S over the whole group to obtain theorthogonality relation

∑

S

Dλ(S−1)klDµ(S)ij = δλµδkjδil

h

nλ(A.9)

where (nλ, nµ) are the respective dimensions of both representations. If theyare unitary, Eq. (A.9) reads

∑

S

Dλ(S)†klDµ(S)ij = δλµδkjδil

h

nλ(A.10)

which is called the fundamental orthogonality relation of the first kind. If wetake i = j, k = l and we sum over i and k we find an orthogonality relation forcharacters of irreducible representations:

∑

S

χ∗λ(S)χµ(S) = hδλµ (A.11)

If we consider an arbitrary representation D we can obtain its decompositioninto irreducible components:

D =∑

λ

cλDλ , cλ = integers ≥ 0 (A.12)

Upon applying relation (A.11) we have a formula for cλ :

cλ =1

h

∑

S

χ∗λ(S)χD(S) (A.13)

cλ is the number of times that the representation λ is contained in D. Thisdecomposition is unique, except for unitary transformations: if two represen-tations yield the same set cλ they can only differ by a unitary transformation.Two representations related by a unitary transformation are called equivalent.

A.4 Projection operators

Given a unitary space R invariant under a group G we can project it onto allthe subspaces it contains which are irreducible under G. We obtain a basis forthe irreducible representation λ as the set of vectors ri, i = 1, · · · , nλ definedas

rλi =

∑

S

Dλ(S)†ijSr (A.14)

A.5. COORDINATE TRANSFORMATIONS 363

where nλ = dimension of Dλ, j is fixed but arbitrary and r is any vector of thespace R. Of course it may happen that the result of applying the projectionoperator

P λi =

∑

S

Dλ(S)†ijS

on r be zero , in which case one has to choose another vector to operate upon.After acting with P λ

i for i = 1, · · · , nλ on one or more (if necessary) vectors rone obtains a basis for the representation λ.

A.5 Coordinate transformations

A change of coordinates, in general a composition of a translation and a ro-tation of the laboratory coordinate system (affine transformation), changes inprinciple the values of all the physical variables. If there is a set of such trans-formations that leave the Hamiltonian invariant (symmetry transformations),they constitute a group, since H is invariant under any composition of two ormore symmetry transformations, the identity is obviously contained in the setand all rotations and translations have an inverse. The transformations can alsoinclude the inversion at a fixed point or reflections on symmetry planes. If weleave translations aside, the symmetry group is called a point group, since itcontains rotations or reflections which leave one point fixed, which we choose asthe origin. Consider now the stationary Schrodinger equation for an ion

H(x)ψiλ(x) = Eλψ

iλ(x) (A.15)

The index i = 1, · · · , nλ denotes one of the linearly independent wave functionsbelonging to the same eigenvalue Eλ with degeneracy nλ. We apply now a trans-formation of the coordinates, of the form x = Rx′, which leaves H invariant, sothat:

H(x)ψiλ(Rx) = Eλψ

iλ(Rx) (A.16)

Changing the coordinates, however, does change the wave function, in general,and we can write

ψiλ(Rx) = (PRψ

iλ)(x) (A.17)

which is a different function of x. This function, since it satisfies the sameeigenvalue equation, must be a linear combination of the basis of the degeneratemanifold of Eλ.Then we write

ψiλ(Rx) =

∑

j

Djiλ (R)ψj

λ(x) (A.18)

It is clear that the matrices D(R) in Eq. (A.18) form a representation of thesymmetry group G of the Hamiltonian. Before we conclude that this is an ir-reducible representation we must discard the possibility that the origin of thedegeneracy be other than symmetry. In fact it is possible that there exist acci-dental degeneracies due to special properties of the potential, which would imply


that the degenerate manifold contains more than one irreducible representationof G. If this situation is discarded, we have the

Theorem A.2 The manifold of nλ linearly independent degenerate eigenfunc-tions of a given eigenvalue Eλ constitutes a basis for an irreducible representa-tion of the symmetry group of H.

A.6 Wigner Eckart theorem

In order to prove Wigner-Eckart theorem we shall make use of a very importantlemma on linear transformations:

Lemma A.1 (Schur) Given D(1)(T ),D(2)(T ), ∀T ∈ G, two irreducible repre-sentations of dimensions n1, n2 and a matrix P such that

PD(1)(T ) = D(2)(T )P , ∀T ∈ G (A.19)

then:

1. if D(1) is not equivalent to D(2), P = 0;

2. if D(1) and D(2) are equivalent, either P = 0 or detP 6= 0;

3. if D(1)(T ) = D(2)(T ), ∀T ∈ G, either P = 0 or P = λ1.

A given irreducible representation Dµ can appear several, or even an infinitenumber of times in the spectrum of a Hamiltonian H with symmetry groupG. Let us call the different equivalent representations Dµr , where we shallassume that we choose always bases such that the matrices for the same µare not only equivalent but in fact identical. We are looking for a generalselection rule for matrix elements. Let us start by considering a set of functionsφλ

j , j = 1, · · · , nλ which transform according to Dλ under G, and expandingthem in terms of the eigenfunctions of H , which form a complete basis set ofthe Hilbert space:

φ(λ)j =

∑

µr

M(µr)ij ψ

(µr)i (A.20)

where

M(µr)ij =

∫

ψ(µr)∗i φλ

j d3r (A.21)

If T ∈ G, Tφ(λ)j can be wrtten in two ways, according to whether we choose or

not to expand φ in terms of ψi:

Tφ(λ)j =

∑

µr

M(µr)ij D

(λ)ki (T )ψ

(µr)k

Tφ(λ)j =

∑

i

D(λ)ij (T )φ

(λ)i

=∑

µr

∑

ik

D(λ)ij (T )M

(µr)ki ψ

(µr)k (A.22)

A.7. SPACE GROUPS 365

Since the l. h. s. is the same in these equations, we have

D(µ)M(µr) = M(µr)D(λ) , ∀T ∈ G. (A.23)

We are in the conditions stated as the hypotheses for Schur’s lemma with M inthe role of the lemma’s P, so that the conclusions are:

Theorem A.3 (Wigner Eckart)

1. When λ 6= µ , M = 0: there cannot be contributions to the expansion(A.20) from functions which do not transform according to Dλ;

2. When λ = µ , M = aλr1: inside each particular set of basis functionsψλr

i the matrix M is a multiple of the unit matrix. The constant ofproportionality depends on the representation and on the particular degen-erate eigenvalue r.

A.7 Space groups

A crystal lattice is invariant under a group S of symmetry operations R | tnwhich are compositions of point group transformationsR and lattice translationstn. S is called the space group of the lattice. The point group operations canbe either pure rotations, called “proper rotations”, or “improper rotations”, thelatter being the product of a proper rotation times the inversion at a latticepoint.

The translations which leave a lattice of d dimensions invariant have theform

tn =

d∑

i=1

niai (A.24)

where ai is a primitive vector basis of the lattice and ni are integers. Infact the infinite lattice itself can be defined as the set of points generated bysubstituting all the integers for the ni in Eq. (A.24). The primitive basis inEq. (A.24) is not unique, although there is always a way of defining the simplestone.

The set of the translations by the vectors tn is a group T, the translationgroup of the lattice, which is in all cases a subgroup of S.

Under a point group transformation one primitive translation is transformedinto a new translation:

t′ = R(tn) (A.25)

which must again have the form (A.24), so that ni → n′i. This is a very

strong restriction on the possible point operations. The main result in this con-nection is that the only proper or improper rotations compatible with a latticein the above sense are integral multiples of π/2 or π/3.


In some cases, the space group operations are such that the point and trans-lation parts do not exist separately as symmetry operations, but their compo-sition does. In such cases the translation is not contained in the set (A.24).These are called non-symmorphic space groups. In a symmorphic space groupthe point symmetry operators constitute a group, which is a subgroup P of thespace group S. A symmorphic space group is the direct product of the point andtranslation subgroups:

S = P ⊗ T .

In non-symmorphic groups some elements of S have the form R | vR, withvR 6= tn (we remark that 0 ∈ tn). Since E | 0 ∈ S, the point operator Rabove 6= E. Then there is a cyclic subgroup of S of order m = integer > 0 and

R | vRm = E | 0 .

Therefore, vR is a rational fraction (q/m)tn , q/m < 1, of a basic translation.The possible primitive bases that fill the space in 3d with points of the form

(A.24) are 14 primitive triads, which generate an equal number of so calledBravais lattices. All these restrictions limit the number of the possible spacegroups to 2 in 1d, 17 in 2d and 230 in 3d. There are 73 symmorphic spacegroups in 3d, the rest of the 230 being non-symmorphic.

Finally, there are only 32 point groups compatible with crystal lattices. Toeach of them corresponds a family of crystals called a “crystalline class”.

A.8 Bloch’s theorem

We consider now specifically the translation subgroup T of a space group S. Tis a commutative group, because the composition of translations is equivalentto an algebraic addition, so that T is isomorphous to the commutative group ofaddition of numbers.Commutative groups are also called Abelian. They have only one dimensional(scalar) irreducible representations (reps). Call χ(t) the character of the trans-lation t for a given rep. In a 1d representation, the matrix D(t) = χ(t).

We must have χ(−t) = χ−1(t) and χ(Nt) = χ(t)N . It is clear that

χ(t) = eik·t

satisfies both conditions, with any k.Consider now a discrete Hamiltonian defined on the sites of a periodic infinite

lattice. A solution of the stationary Schrodinger equation

Hψα = Eαψα (A.26)

must be a basis for a rep of the translation group T, since H is invariant underthat group. Since all reps of T are 1d, we have

HD(tn)ψα = Hχα(tn)ψα = Eαχα(tn)ψα (A.27)

A.8. BLOCH’S THEOREM 367

We already found χα(tn) = eik·tn so that we can use k as a label of the eigen-solution, and write α = (k, µ), µ being the remaining set of quantum numbersnecessary to characterize the eigenvalue. The wave function satisfies

D(tn)ψα = eik·tnψα (A.28)

Clearly the form

ψα = ψkµ = A∑

n

eik·tnφµ(Rn) (A.29)

satisfies (A.28). Eq. (A.29) is Bloch’s theorem for this case.For an electron in a crystalline periodic potential the stationary Schrodinger

equation readsH(r)ψα(r) = Eαψα(r) (A.30)

Performing a lattice translation leaves H invariant:

H(r)ψα(r + tn) = Eαψα(r + tn) (A.31)

Repeating the argument above, we find that α must contain a given wave-vectork and other quantum numbers not related to translational symmetry, so that

ψα(r) = eik·rukµ(r) (A.32)

where ukµ(r + tn) = ukµ(r) is periodic. This form automatically satisfies

ψkµ(r + tn) = eik·tnψkµ(r) (A.33)

Eq. (A.32) is Bloch’s theorem for this case.

Appendix B

Time Reversal

B.1 Antilinear operators

We shall always consider linear or anti-linear operators acting on a Hilbert vectorspace. Let us first enunciate two useful theorems.

Theorem B.1 Two linear operators A,B are equal iff

〈 u | A | u 〉 = 〈 u | B | u 〉 ∀ | u 〉 (B.1)

Theorem B.2 Two linear operators A,B are equal up to a phase factor

A = Beiα (B.2)

iff|〈 u | A | v 〉| = |〈 u | B | v 〉| ∀ | u 〉, | v 〉 (B.3)

Definition B.1 A is an anti-linear operator if for any pair of kets | u 〉 , | v 〉 ,and any pair of complex numbers λ , µ

A (λ | u 〉 + µ | v 〉) = λ∗A | u 〉 + µ∗A | v 〉 (B.4)

In particular we observe that an anti-linear operator does not commute with ac-number:

A (c | u 〉) = c∗A | u 〉 (B.5)

We recall the definition of the hermitian adjoint of a linear operator A. Firstconsider the action of A on a bra. Let us take any bra 〈χ |. The scalar product〈χ | (A | u〉) is a linear function of | u〉 because A = linear. We know that alinear function of the vectors (kets) belonging to a Hilbert space defines a vector(bra) of the dual vector space, so we call this bra 〈η |. There is a one-to-onelinear correspondence between 〈χ | and 〈η |, because the scalar product is alsoa linear operation. We then define:

〈η |≡ 〈χ | A (B.6)

369

370 APPENDIX B. TIME REVERSAL

and we have the identity

(〈χ | A) | u〉 = 〈χ | (A | u〉) (B.7)

Let us take now the ket | v〉 = (〈u | A)†. This is an anti-linear function of the

bra 〈u |:(

∑

i

λi〈ui | A)†

=∑

i

λ∗i | vi〉 (B.8)

But

(

∑

i

λi | ui〉)†

A

†

=

(

∑

i

λ∗i 〈ui |)

A

†

=

(

∑

i

λ∗i 〈vi |)†

=∑

i

λi | vi〉 (B.9)

is a linear function of | u〉, which we shall define as the operator A†:

Definition B.2 A†is the hermitian adjoint (h. a.) of A if

A† | u 〉 = (〈 u | A)†, ∀ | u 〉 (B.10)

Let us now consider the effect an anti-linear operator has on a bra. If A isanti-linear and 〈 χ | is any bra, the scalar product

〈 χ | (A | u 〉)

defines an anti-linear function of the ket | u 〉. Therefore, its complex conjugateis a linear function, which defines a bra η such that

〈 χ | (A | u 〉)∗ ≡ 〈 η | u 〉 ≡ (〈 χ | A) | u 〉 (B.11)

We have obtained the general identity, for anti-linear operators:

(〈v | A) | u〉 ≡ 〈v | (A | u〉)∗ (B.12)

This defines an anti-linear function of the bra 〈 χ |:(

∑

i

λi〈 χi |)

A =∑

i

λ∗i 〈 χi | A (B.13)

If we use definition (B.10) for the h. a. of any operator

A† | u〉 ≡ (〈u | A)† (B.14)

B.2. ANTI-UNITARY OPERATORS 371

we find that for an anti-linear A the h. a. satisfies the relation

〈t |(

A† | u〉)

= 〈u (A | t〉) (B.15)

to be compared with the corresponding relation for a linear operator:

〈t |(

A† | u〉)

= 〈u (A | t〉)∗ (B.16)

Clearly if A is anti-linear A† is also anti-linear.

B.2 Anti-unitary operators

Definition B.3 An anti-linear operator A is anti-unitary if its h. a. equals itsinverse:

AA† = A†A = 1

An anti-unitary operator V induces a mapping

| u 〉 = V | u 〉 (B.17)

An operator B is transformed by V as

B = V BV † (B.18)

and a bra as〈 v |= 〈 v | V † (B.19)

Since anti-unitary operators are anti-linear, we have

〈 η | (V | v 〉) = 〈 v |(

V † | v 〉)

(B.20)

which leads to the relation

〈 u | B | v 〉 = 〈 u | B | v 〉∗ (B.21)

so that matrix elements are not invariant under V , but are transformed insteadinto their complex conjugates.

B.3 Time reversal

Equation (B.4) implies that an anti-unitary operator transforms a c-numberinto its complex conjugate:

AcA† = c∗AA† = c∗ (B.22)

so that, in particular, the basic commutators among dynamical operators changesign:

[ p, q ] = ih =⇒ [ p, q ] = −ih (B.23)

372 APPENDIX B. TIME REVERSAL

Let us call K0 an operator which when applied on the coordinate and linearmomentum operators yields

K0rK†0 = r

K0pK†0 = −p (B.24)

This operator clearly changes the sign of the commutator, and its square is theidentity. It is easy to see that it is anti-unitary. One can prove that K†

0 = K0 =K−1

0 . On the other hand we observe that its action on the coordinates andmomenta, and accordingly on the angular momentum, coincides with the effectof reversing the sign of time, so we take K0 as the time-reversal operator. Inorder to completely define K0 we may demand that when it is applied on thekets of the basis set of eigenfunctions of H it leaves them invariant:

K0 | n 〉 =| n 〉 (B.25)

which, when combined with its anti-linear properties implies that the coordi-nates in the basic set (that is, the wave function) of an arbitrary ket 〈 n | u 〉 istransformed by K0 into its complex conjugate:

〈 n | (K0 | u 〉) =∑

m

〈 n | (K0 | m 〉〈 m | u 〉) =

〈 n | u 〉∗ (B.26)

Also, the representation matrix elements of any linear operator are transformedby K0 into their complex conjugates. In effect, we have seen that for an anti-unitary operator,

〈 v | (A | u 〉) = (〈 v | A) | u 〉 ∗

Then

〈 m | B | n 〉 = 〈 m | K0BK0 | n 〉 = 〈 m | (K0B | n 〉) =

[ (〈 m | K0)B | n 〉 ]∗

= 〈 m | B | n 〉∗ (B.27)

for any operator B.If the Hamiltonian is invariant under time-reversal, we have

[ K0, H ] = 0 . (B.28)

Consider now the time-dependent Schrodinger equation

ih∂ | Ψ(t)〉

∂t= H | Ψ(t)〉 (B.29)

Applying K0 on both sides we find that the state

| Ψrev(t)〉 = K0 | Ψ(−t)〉

B.3. TIME REVERSAL 373

satisfies the same equation B.29. In particular, consider an eigenstate | φ〉 whichsatisfies the stationary Schrodinger equation

H | φ〉 = E | φ〉 (B.30)

with E = real, and apply K0 on both sides. We find, taking into account that[H,K0] = 0 and that we have chosen the representation in which eigenkets arereal, that the wave function

φm = 〈m | φ〉 (B.31)

and its complex conjugate φ∗m are degenerate. In the coordinate representation,

this implies that if Ψ(r) is an eigenfunction of (B.30) then Ψ∗(r) is anothersolution with the same energy eigenvalue. Therefore, they are either propor-tional to each other or linearly independent; in the latter case they span atwo-dimensional degenerate subspace of that particular eigenvalue.

Index

adiabatic switching, 153Anderson Hamiltonian, 263angular momentum operators, 79antiferromagnet phase transitions, 68antiferromagnet wih exchange and

dipolar forces, 289antiferromagnet, longitudinal suscep-

tibility, 63antiferromagnet, longitudinal suscep-

tibility in MFA, 60antiferromagnet, longitudinal suscep-

tibility in RPA, 148antiferromagnet, transverse suscep-

tibility in RPA, 149antiferromagnet, transverse suscep-

tibility, 65antiferromagnetism in MFA, 61atomic configuration, 6

band structure effects on indirect ex-change, 248

bilayer with exchange and dipole in-teractions, 298

bipartite lattices, 61Bloch’s theorem, 81Bogoliubov inequality, 278Bogoliubov transformation, 94Boltzmann equation, 268Born theory of scattering, 268Brillouin function, 3, 9

Clebsch-Gordan coefficients, 6closure property of coherent states,

185coherent state, 186collision-time, 269complex magnetic susceptibility, 180

correlation function, 131correlation length, 132correlation length, antiferromagnet,

140critical behaviour of ferromagnet, 46critical exponent, 29crystal-field, 4, 11cubic symmetry , 12Curie law, 10Curie temperature, 29Curie’s law, 2Curie- Weiss temperature, 45Curie-Weiss law, 234Curie-Weiss law, 29

demagnetization factors, 176density matrix, 153Devlin’s method, 151dilation, 111dipolar interaction in layers, 295dipolar interactions, 29dipolar interactions in low dimen-

sions, 282dipolar surface effects, 323dipole-exchange cross-over, 286Dzialoshinsky-Moriya interaction, 107

effective magneton number, 4elastic constants, 113elastic energy, 109electron correlations, 222ellipticity of spin precession, 176, 177Ewald’s lattice summation method,

284exchange gap, 230exchange interactions, 29

375

376 INDEX

ferromagnetic metals with degener-ate bands, 210

ferromagnetic relaxation, 346Feynmann-Helmann theorem, 229fluctuation-dissipation theorem, 120,

230FM with finite applied field, 135formal scattering theory, 269free energy, 232free energy, 149, 230

Golstone’s theorem, 206Green’s functions, 264Green’s functions definition, 117Green’s functions spectral represen-

tation, 120Green’s functions, analytic proper-

ties, 120group, definition, 360groupoid, 359

Hartree–Fock approximation, 198, 264Heisenberg antiferromagnet, 92Heisenberg ferromagnet, 79Heisenberg Hamiltonian, 29, 36Heisenberg picture, 153Heitler-London approximation, 30helimagnetism, 60, 72Helmholtz free energy, 43, 49Holstein-Primakoff transformation, 82Hubbard Hamiltonian, 198Hund’s rules, 8hysteresis of the Weiss model, 49

indirect exchange, 40, 247indirect exchange in semiconductors,

256interaction picture, 154inverse magnetostriction, 113iron group, 10Ising model, 288

Kosterlitz-Thouless phase transition,321

Kramers-Kronig relations, 120, 121Kubo formula, 153

Lande g factor, 9Langevin function, 2linear FM chain, 132linear response theory, 153longitudinal susceptibility, 29

magnetic impurities, 263magnetization curve in MFA, 47magneto-crystalline anisotropy, 101magneto-elastic coupling, 108magneto-elastic energy, 109, 110magnetostatic modes, 179magnetostriction, 111magnon, 81, 84magnon interactions, 84magnon manifold, 175magnon-phonon bilinear interaction,

355magnon-phonon interaction, 345, 353magnons in itinerant systems, 200Mermin-Wagner theorem, 277metamagnet, 70molecular field, 27molecular field approximation, 43monoid, 359monolayer with dipolar and exchange

interactions, 297

Neel temperature, 141Nagaoka-Thouless theorem, 223nesting , 214non linear corrections to SWA, 91non-orthogonal set of states, 31

order parameter, 47ordinary, special and extraordinary

surface phase transitions, 308overlap, 31, 33

paramagnons, 228Pauli susceptibility, 195, 214phase states, 188pseudo-dipolar interaction, 106pseudo-quadrupolar interaction, 108

quantum critical point, 228

INDEX 377

quenching of orbital angular momen-tum, 3

resistance minimum, 267resolvent operator, 270RKKY interaction, 40rotation symmetry group, 8RPA for antiferromagnet, 136RPA for spin 1/2 FM, 122RPA for the AFM chain, 291RPA susceptibility of metals, 200RPA, comparison with MFA, 126Ruderman-Kittel interaction, 246Russel-Saunders coupling, 4

s-d Hamiltonian, 263Schrodinger picture, 153Schur’s lemma, 185second order phase transition, 29,

46second quantization, 31, 32, 197semiclassical picture of spin wave,

85semigroup, 359shape-anisotropy energy, 172single ion anisotropy, 104single site anisotropy, 150space groups, 365space quantization, 3specific heat divergence, 229specific heat of ferromagnet, 91spectroscopic term, 5spin-density wave, 213–215spin-flop, 68spin-flop transition, 146spin-orbit coupling, 5, 7square FM lattice, 133staggered field, 136staggered field in antiferromagnet,

62static susceptibility of FM, 129Stevens operator equivalents, 14Stoner excitations, 203

Stoner instability, 198Stoner model of band magnetism,

200strain tensor, 109stress tensor, 113superexchange, 36surface anisotropy, 322surface density of magnon states, 317surface magnetism in the MFA, 306susceptibility, 10susceptibility of ferromagnet in RPA,

158symmorphic space groups, 366

T matrix, 270tesseral harmonics, 15time reversal, 4total spin deviation, ferromagnet, 89transfer matrix, 313two-magnon bound state in 2d, 339two-magnon states in one dimension,

337

uncertainty relations, 187uniform precession frquency, 176

variance, 187

Weiss ferromagnetism, 27Weiss ground state of ferromagnet,

80Weiss model, 45Wigner-Eckart theorem, 8

Zeeman interaction, 8zero point motion in ferromagnet,

86zero-point dipolar spin-deviation, 178zero-point energy of antiferromag-

net, 99zero-point spin deviation of AFM,

139

the quantum theory of magnetism

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