Two-Hole Bound States from an

Effective Field Theory for Magnons

and Holes in an Antiferromagnet on

the Honeycomb Lattice

Masterarbeit

der Philosophisch-naturwissenschaftlichen Fakultat

der Universitat Bern

vorgelegt von

Marcel Wirz

2009

Leiter der Arbeit

Prof. Dr. Uwe-Jens Wiese

Institut fur Theoretische Physik, Universitat Bern

Abstract

In this thesis a systematic low-energy effective field theory for magnons and holes in an anti-ferromagnet on the honeycomb lattice is constructed analogous to baryon chiral perturbationtheory (BχPT). The Hubbard model serves as an underlying microscopic model. We showthat the leading action of the pure magnon sector is the same as in the square lattice case.Using the effective theory, we derive the one-magnon exchange potentials. With these po-tentials, we investigate the Schrodinger equation describing the relative motion of two holesin an otherwise undoped antiferromagnet. We derive the radial solution of this Schrodingerequation for a special case analytically and find that the magnon-mediated forces lead totwo-hole bound states under a condition on the low-energy constants.

Also the weak coupling limit of the Hubbard model is investigated. In this context we de-rive the energy-momentum relation in the zero coupling limit, where massless Dirac fermionsarise. In addition to deriving their velocity, we construct the leading terms of the correspond-ing effective Lagrangian.

Contents

1 Introduction 1

2 The Honeycomb Lattice 5

2.1 A Bipartite Non-Bravais Lattice . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Symmetries of the Honeycomb Lattice . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Shift Symmetries Di . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Rotation O and Reflection R Symmetries . . . . . . . . . . . . . . . . 7

3 The underlying Microscopic Models and their Symmetries 9

3.1 The Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 The t-J Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 The Quantum Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 The Strong Coupling Limit of the Hubbard Model and how it leads to Anti-ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 Symmetries of the Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5.1 SU(2)s Spin Rotation Symmetry . . . . . . . . . . . . . . . . . . . . . 11

3.5.2 U(1)Q Fermion Number Symmetry . . . . . . . . . . . . . . . . . . . . 12

3.5.3 Discrete Symmetries Di, O, O′, and R . . . . . . . . . . . . . . . . . . 12

3.5.4 Non-Abelian Extension of the U(1)Q Symmetry to SU(2)Q . . . . . . 13

3.6 Manifestly SU(2)s ⊗ SU(2)Q Invariant Formulation of the Hubbard Model . . 13

4 Investigation of the Free Fermion Model 15

4.1 The Weak Coupling Limit of the Hubbard Model and how it leads to MasslessDirac Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Effective Field Theory for Dirac Fermions . . . . . . . . . . . . . . . . . . . . 19

4.2.1 Fermion Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2.2 Effective Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Magnons 25

5.1 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Effective Theory for Magnons . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.3 Non-linear Realization of the SU(2)s Symmetry . . . . . . . . . . . . . . . . . 28

5.4 Composite Magnon Field vµ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 From Microscopic Operators to Effective Fields for Holes 35

6.1 Fermion Operators with a Sublattice Index . . . . . . . . . . . . . . . . . . . 35

6.2 Fermion Fields with a Sublattice Index . . . . . . . . . . . . . . . . . . . . . . 36

i

ii Contents

6.3 Fermion Fields with a Sublattice and Momentum Index . . . . . . . . . . . . 386.4 Hole Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7 Effective Field Theory for Magnons and Holes 43

7.1 Effective Action for Magnons and Holes . . . . . . . . . . . . . . . . . . . . . 437.2 Accidental Emergent Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.2.1 Galilean Boost Symmetry G . . . . . . . . . . . . . . . . . . . . . . . . 457.2.2 Continuous O(γ) Rotation Symmetry . . . . . . . . . . . . . . . . . . 46

8 One-Magnon Exchange Potentials 47

8.1 One-Magnon Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488.3 Transforming to Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . 498.4 Principle of Stationary Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.5 Potentials in Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . . . 518.6 Potentials in Position Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

9 Two Hole Bound States 55

9.1 Corresponding Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . 55

10 Conclusions and Outlook 59

Acknowledgements 61

Bibliography 63

Chapter 1

Introduction

The ultimate motivation of this thesis is to contribute to the understanding of high tem-perature superconductivity. Ordinary superconductivity, which was discovered in 1911, isdescribed by the Bardeen-Cooper-Schrieffer (BCS) theory. It results from Cooper pair forma-tion of two electrons. An attractive interaction mediated by phonon exchange may overcomethe Coulomb repulsion. The condensation of the Cooper pairs then leads to superconductivity.

High temperature superconductivity was discovered in 1986 by Bednorz and Muller [12].The responsible mechanism is still not known and the BCS theory does not apply, since inordinary superconductors the Cooper pairs are coupled weakly, while in high temperaturesuperconductors (HTS) they are coupled strongly. For high temperature superconductivityone assumes that phonons alone can not provide a mechanism for Cooper pair formation.

Beside new iron-based materials, most known HTS result from hole- (missing electrons)or electron-doped layered cuprates (copper-oxides). Before doping, the systems are antifer-romagnetic insulators. As most of the known materials come with a square lattice structure,we first discuss them. They have a common basic structure. Two-dimensional CuO2 layersare separated by insulating layers. They can be doped by substituting ions in the insulat-ing layers, with the result that the additional electrons or holes reside in the cuprate layers.As first understood by Anderson [23], the CuO2 planes are responsible for high temperaturesuperconductivity. Since the cuprate layers are coupled very weakly, the relevant physics istwo-dimensional and therefore we will work in two dimensions in the following.

In fig. 1.1, the antiferromagnetic and the superconducting phases of an electron- and ahole-doped material on square lattices are shown. For materials with a honeycomb latticeone also finds such antiferromagnetic and superconducting phases. One can see that the an-tiferromagnetic phases only exist at sufficiently low temperatures and at zero or light doping.Caused by very small couplings between the different layers, the antiferromagnetic correlationlength remains infinite at low non-zero temperatures in contrast to the case of an exactly two-dimensional antiferromagnet, which we consider in this thesis. The superconducting phasesonly arise at doping rates at which the antiferromagnetic order has disappeared.

Most of what we discuss in this thesis has already been done for the square lattice case in[1–8]. However, there are also HTS with other lattices, for example with a honeycomb lattice,which we investigate in this thesis. For example, the dehydrated variant of NaxCoO2·yH2Oat x = 1/3 shows honeycomb structure. Its physics is similar to the one of the cuprateswith the difference that there are CoO2 instead of CuO2 layers. In fig. 1.2, its structure isillustrated. Another antiferromagnetic material with a honeycomb lattice is InCu2/3V1/3O3,

1

2 1. Introduction

Figure 1.1: Schematic phase diagram of electron- (left) and hole-doped (right) cuprates, showingantiferromagnetism (AF) and superconductivity (SC). (Figure taken from [40].)

again with CuO2 layers. As actual materials with honeycomb structure are hole doped, werestrict ourselves to the hole doped sector. However, the procedure for the electron dopedsector would be analogous. When we speak about holes in the following, we mean dopedholes relative to a half-filled system (one electron per lattice site on average).

Figure 1.2: Structural views of Na0.7CoO2 (left) and NaxCoO2·yH2O (right), where Na andH2O sites are partially occupied. (Figure take from [39].)

As two-dimensional quantum antiferromagnets are not yet completely understood evenbefore they turn into HTS upon doping, we investigate their low-energy physics in this thesisfor materials with a honeycomb lattice. There exist minimal microscopic models like theHubbard or the t-J model, which describe their physics. It is probably not possible to solvethese models analytically and except for the undoped, the one-hole, and the one-electronsector, they can not even be solved numerically because of a severe fermion sign problem.Also the exact ground state is not known. Therefore we proceed in another way and constructa systematic low-energy effective field theory for magnons and holes in an antiferromagnet

3

on the honeycomb lattice analogous to baryon chiral perturbation theory (BχPT) in thisthesis. Then we use this theory to investigate the Schrodinger equation for the relativemotion of two holes. Also other applications like spirals in the staggered magnetization canbe investigated [9, 11].

A characteristic feature of quantum antiferromagnets is the spontaneous symmetry break-ing of the global SU(2)s spin rotation symmetry down to the subgroup U(1)s. According toGoldstone’s theorem, the result are two Goldstone bosons, the magnons or antiferromag-netic spin waves. Because they are massless, the low-energy physics is dominated by theirdynamics.

In the antiferromagnetic phase, the fermion number Q is a conserved quantity. This,together with the spontaneously broken SU(2)s spin symmetry, allows the construction of alow-energy effective field theory for magnons and holes analogous to BχPT. BχPT and theeffective field theory for magnons and holes are compared in [1–4,7].

In Quantum Chromodynamics (QCD), which describes the strong interactions of quarksand gluons, one is also confronted with the problem that it is probably not analyticallysolvable. Therefore, a systematic effective field theory, BχPT was developed, which describesthe low-energy physics of QCD.

Spontaneous symmetry breaking also occurs in QCD and results in three Goldstonebosons, the pions π+, π0, π−, which dominate the low-energy physics of QCD. Since thebaryon number B is conserved in QCD, each baryon sector can be investigated separately.For the pure pionic sector (B = 0), a low-energy effective field theory known as chiral pertur-bation theory (χPT) was developed in [14, 17]. For the case B 6= 0 the theory was extendedto baryon chiral perturbation theory (BχPT) [18–22].

The effective theory for magnons and holes is used to investigate the low-energy physicsof antiferromagnets. Undoped antiferromagnets are governed by magnons. Doped antifer-romagnets are of particular interest. At low energy, the dynamics of holes is dominated bylong-range magnon-mediated forces. In this thesis we investigate the leading process, one-magnon exchange. Since HTS result from doped antiferromagnets, processes like magnonexchange might be important for Cooper pair formation.

One may ask whether the effective theory for magnons and holes is applicable to HTS.Since the antiferromagnetism has disappeared, the SU(2)s symmetry is no longer brokenspontaneously. Therefore the perturbative treatment of the effective theory breaks down.However, the effective theory itself does not as long as the magnons, now with a finite corre-lation length, and the holes with the index structure of the antiferromagnetic phase, remainamong the relevant degrees of freedom. In this thesis we limit ourselves to perturbativecalculations in the antiferromagnetic phase.

We will now describe what a low-energy effective field theory is. Most microscopic mod-els which describe the physics of a system up to high energies can probably not be solvedanalytically. What one can do is to catch the relevant degrees of freedom at low energies anddescribe only those. Since such a theory contains only the relevant low-energy physics, it iseasier to handle, and therefore called effective. Of course, such an effective theory is onlyvalid up to a certain energy scale, but below it it is equivalent to the underlying microscopictheory.

For the construction of an effective theory one first has to work out all symmetries of theunderlying theory. Then all relevant degrees of freedom at low energies and their transforma-tion behaviour under the various symmetries must be worked out. At last, as Weinberg statedin [14], one has to construct the most general Lagrangian consisting of all terms which are

4 1. Introduction

invariant under all symmetries. These terms consist of the appropriate field degrees of free-dom and their derivatives. Derivatives correspond to momenta. Therefore, the Lagrangian ofthe effective theory is constructed as a derivative expansion, where with each order furthercorrections are taken into account. Since we are interested in the low-energy physics, we onlyhave to construct the leading orders, i.e. terms with a small number of derivatives. Thus, thegoal is to write down the leading terms of the Euclidean action, used in a path integral.

Each term is endowed with a low-energy constant which determines the coupling strengthof the corresponding interaction. The numerical values of these low-energy constants cannot be determined in the framework of the effective theory, but have to be determined byexperiment or numerical simulations. The high-energy physics of the underlying model isabsorbed in these low-energy constants. Since short distances correspond to high momenta,the effective theory is insensitive to the microscopic details of the underlying model. Thus, theeffective theory is the same for all underlying models sharing the same symmetries. Differentmodels just have different values of the low-energy constants.

Besides fundamental principles of field theory like locality and unitarity, the effective the-ory for magnons and holes constructed in this thesis relies on three inputs. First, the Hubbardmodel serves as an underlying microscopic model. (As mentioned, except for the numericalvalues of the low-energy constants, the effective theory remains the same for all microscopicmodels which have the same symmetries as the Hubbard model.) The second input is thatthe global SU(2)s spin symmetry is spontaneously broken down to U(1)s. And the thirdinput is that the holes reside in momentum space pockets centered around (0, 4π/(3

√3a))

and (0,−4π/(3√

3a)) where a is the spacing between two neighbouring lattice sites. The restof the construction is analogous to BχPT. Apart form these inputs, the construction is onlybased on symmetries, which the effective theory inherits from the underlying model. Thisshows the strength and importance of symmetries.

The effective theory is only valid for low energies. In the context of the Hubbard model,low energies are small compared to the parameters t and U which set the energy scales in thismodel. These energy parameters are incorporated in the low-energy constants of the effectivetheory. Thus, low energy in the context of the effective theory means small compared to alow-energy constant like the spin stiffness ρs.

In the microscopic theory, the electrons are coupled strongly. What makes the effec-tive theory so powerful, is the fact that the holes are derivatively coupled to the magnons.Therefore, the long-range forces mediated by magnon exchange between the holes can beinvestigated using perturbation theory.

Since the microscopic models are defined on a lattice, the Poincare-invariance is explicitlybroken and so the underlying models and with them the effective theory are nonrelativistic. Inaddition, because of this explicit symmetry breaking, the models and the effective theory donot contain phonons. In real materials phonons are the Goldstone bosons of the spontaneouslybroken symmetries caused by the formation of the crystal lattice.

I have done this Master thesis in close collaboration with Banz Bessire who wrote his thesisabout the same effective theory, however, with applications to spiral phases of the staggeredmagnetization [9]. For the square lattice case, such an effective theory has been developedand also the two-hole bound states as well as the spiral phases have been investigated. Weused the corresponding theses and papers and adapted everything to the honeycomb lattice.Thus, this thesis closely follows especially [1] as well as [2–8]. We work in natural units, i.e.~ = 1 in the whole thesis.

Chapter 2

The Honeycomb Lattice

In this thesis we investigate strongly correlated electron systems on an infinitely extendedhoneycomb lattice with a lattice spacing a between two neighbouring sites. As mentioned inthe introduction, the effective theory will inherit all symmetries of the underlying model andwith this also the ones of the honeycomb lattice. This chapter discusses some important factsconcerning the honeycomb lattice and its symmetry properties.

2.1 A Bipartite Non-Bravais Lattice

The honeycomb lattice is bipartite, which means that it can be divided into two sublatticesA and B such that all sites of A only have nearest neighbours on B and vice versa. Thiswill become important later as symmetry transformations can map each sublattice on itselfor A on B and vice versa. The two sublattices are illustrated in fig. 2.1. Every site has threenearest neighbours, thus the coordination number is three, which differs from the coordinationnumber of the square lattice, which is four. This fact explains the difference of some low-energy constants, see section 5.2.

Later we will become interested in the first Brillouin zone. This can only be constructedfrom a Bravais lattice, which the honeycomb lattice is not. A definition of a Bravais latticecan be found for example in [42] and reads: ”A Bravais lattice is an infinite array of discretepoints with an arrangement and orientation that appears exactly the same, from whicheverof the points the array is viewed.” Taking a look at fig. 2.1, it is clear that the points onsublattice A look different than those on B. However, the two triangular sublattices A andB are Bravais lattices and we can construct a Brillouin zone for each of them. This is whatwe will do now. Both sublattices are spanned by the primitive lattice vectors

a1 =(

32a,

√3

2 a)

, a2 =(

0,√

3a)

, (2.1)

and with the relationgi · aj = 2πδij , (2.2)

we can calculate the primitive lattice vectors of the reciprocal lattices

g1 =(

4π3a , 0

)

, g2 =(

−2π3a ,

2π√3a

)

. (2.3)

As illustrated in fig. 2.2, we see that the reciprocal sublattices are again triangular, butrotated by 30 degrees with respect to the ones in position space. The figure also shows the

5

6 2. The Honeycomb Lattice

a1

a2

x1

x2

x1

x2

Figure 2.1: The sublattices A (represented by disks), B (represented by circles), and the translationvectors a1, a2.

first Brillouin zone, which is a regular hexagon. For the first Brillouin zone of the honeycomblattice this means that it looks like the one of a triangular sublattice, but it is doubly-coveredas we have one Brillouin zone for the A and one for the B sublattice. In the following A andB will appear as quantum numbers in addition to spin and momentum. Hence, a momentumstate may be occupied by an electron with spin up from lattice A, one from B and with aspin down electron from A and one from B.

g1

g2

Figure 2.2: Reciprocal sublattice and its first Brillouin zone.

An electron or a hole with a momentum k exactly behaves like one with momentum k′,when k and k′ only differ by a lattice vector of the reciprocal lattice. For physical quantitieswhich depend on the momentum, this means that only the values in the first Brillouin zone areneeded. This periodicity in momentum space implies that for such quantities there are onlytwo corners of the first Brillouin zone of the honeycomb lattice which are different. Whichones are equivalent is shown in fig. 2.3. The coordinates of the first Brillouin zone’s corners

2.2 Symmetries of the Honeycomb Lattice 7

are

k1 =(

2π3a ,

2π3√

3a

)

, k2 =(

0, 4π3√

3a

)

, k3 =(

−2π3a ,

2π3√

3a

)

,

k4 =(

−2π3a ,− 2π

3√

3a

)

, k5 =(

0,− 4π3√

3a

)

, k6 =(

2π3a ,− 2π

3√

3a

)

. (2.4)

And its area is given by

ABZ =8√

3π2

9a2. (2.5)

k2

k1k3

k6k4

k5

Figure 2.3: Equivalent corners of the first Brillouin zone.

2.2 Symmetries of the Honeycomb Lattice

Later, when constructing the effective theory, we will search terms which respect all sym-metries of the underlying model. First of all, we are interested in a minimal generatingsystem of all symmetry transformations, and now we are looking for such a system includingall symmetries of the lattice. As mentioned before, it will become important whether thetransformations map the sublattices on themselves or A is mapped on B and vice versa.

2.2.1 Shift Symmetries Di

Looking at fig. 2.1, we see that a minimal generating system of shift symmetries is given bythe drawn translation vectors

a1 =(

32a,

√3

2 a)

, a2 =(

0,√

3a)

. (2.6)

We call the vectors a1 and a2 and the transformations D1 and D2. In contrast to the squarelattice, where the shift symmetries map A on B and vice versa, here the sublattices aremapped onto themselves.

2.2.2 Rotation O and Reflection R Symmetries

The six rotations by a multiple of 60 degrees around a center of a hexagon leave the latticeinvariant. In addition, the three lines going through the edges of a hexagon and the threeperpendicular bisectors of a hexagon side can be viewed as reflection axes. Point reflections at

8 2. The Honeycomb Lattice

a midpoint of a hexagon are the same as a reflection at an axis parallel to the x1-axis and oneparallel to the x2-axis, thus they don’t have to be considered. These six rotations and six re-flexions form the so called dihedral group D6. From this we know that the minimal generatingsystem consists of the rotation of 60 degrees and the reflexion at the x1-axis. Together with theshift transformations we have a minimal generating system which covers all symmetries. Therotation O turns a lattice point x = (x1, x2) into Ox = (1/2x1 −

√3/2x2,

√3/2x1 + 1/2x2)

and the reflection R turns it into Rx = (x1,−x2). Again in contrast to the square lattice,where the rotation maps the sublattices onto themselves, here the A lattice is mapped to theB and vice versa.

Chapter 3

The underlying Microscopic Models

and their Symmetries

As in BχPT, our effective field theory inherits the symmetries from the underlying model.In this chapter we introduce the Hubbard model, which serves as a concrete underlyingmicroscopic model for a quantum antiferromagnet. It shares the important symmetries withthe actual materials and is supposed to contain the relevant physics. We also introduce thet-J and the Heisenberg model as limiting cases of the Hubbard model. By working with suchmodels, we populate the lattice with electrons. First we briefly introduce the three modelsand explain some important facts. Then we perform a detailed symmetry analysis of theHubbard model, as this serves as underlying model for the effective theory.

3.1 The Hubbard Model

We first write down the second quantized Hamiltonian which defines the Hubbard model

H = −t∑

〈xy〉

(

c†xcy + c†ycx)

+U

2

∑

x

(

c†xcx − 1)2

− µ∑

x

(

c†xcx − 1)

, (3.1)

where 〈xy〉 denotes a pair of nearest neighbour sites x and y. The spins at a site x = (x1, x2)

are represented by fermion creation operators c†xs and annihilation operators cxs in spinornotation

c†x =(

c†x↑, c†x↓

)

, cx =

(

cx↑cx↓

)

, (3.2)

whose components obey the standard anticommutation relations, which incorporate the Pauliprinciple

cxs†, cys′ = δxyδss′ , cxs, cys′ = c†xs, c†ys′ = 0. (3.3)

Spin up or down is indicated by arrows (↑, ↓) or s and s′, respectively. This model wasintroduced to describe the transition from conducting to insulating behaviour of stronglycorrelated materials. It has the simplest Hamiltonian which describes mobile interactingelectrons on a lattice and includes two competing processes: the hopping of the electronsbetween nearest neighbour lattice sites (if not forbidden by the Pauli principle) and thelocalization of the electrons on the sites, the latter caused by the Coulomb repulsion betweenthe electrons. Each hop costs a certain amount of energy, which is determined by the hopping

9

10 3. The underlying Microscopic Models and their Symmetries

parameter t. The interaction between the electrons is simply described by the on-site Coulombrepulsion. If two electrons sit on the same site, this costs an energy U > 0. In addition, µ isthe chemical potential for fermion number relative to half-filling (on average one electron perlattice site, µ = 0) which allows us to dope the system with electrons or holes.

3.2 The t-J Model

Another simple description of mobile interacting electrons which is supposed to describequantum antiferromagnets, is the t-J model. It is defined by the Hamiltonian

H = P

− t∑

〈xy〉

(

c†xcy + c†ycx)

+ J∑

〈xy〉

~Sx · ~Sy − µ∑

x

(

c†xcx − 1)

P. (3.4)

The spin operator at the lattice site x is given by

~Sx = c†x~σ

2cx, (3.5)

where ~σ are the Pauli matrices. As in the Hubbard model, t is the hopping parameter, µ isthe chemical potential for fermion number relative to half-filling, and J describes the couplingbetween neighbouring spins. Also in this model only nearest neighbour interactions are takeninto account. P is a projection operator which restricts the Hilbert space by eliminatingdoubly occupied sites, such that only singly occupied or empty sites are allowed. Thereforethe t-J model can solely be doped with holes, while the Hubbard model describes both, theelectron and the hole sector. It has the same symmetries as the Hubbard model except fora non-Abelian SU(2)Q extension of the fermion number symmetry (which we will introducelater). This symmetry is not present in the t-J model, because it describes only the holedoped sector. During the construction of the effective theory this additional symmetry ofthe Hubbard model is broken, such that in both cases we have the same set of symmetries.In [10], for the simulations of mobile holes for the single hole sector, the t-J model is used.

3.3 The Quantum Heisenberg Model

The quantum Heisenberg model describes magnetism in spin systems on a lattice. In contrastto the previous models, the spins are fixed to a specific site. It has the simplest Hamiltoniandescribing spin interactions which is defined by

H = J∑

〈xy〉

~Sx · ~Sy, (3.6)

where J is again the exchange coupling constant. The spin operator at site x is given by

~Sx =1

2~σx. (3.7)

Whether the sign of J is positive or negative defines if parallel or antiparallel spin alignmentis favored. We are interested in antiferromagnetism, thus J > 0.

3.4 The Strong Coupling Limit of the Hubbard Model and how it leads to Antiferromagnetism 11

3.4 The Strong Coupling Limit of the Hubbard Model and

how it leads to Antiferromagnetism

In the strong coupling limit (U ≫ t) for µ 6= 0, the Hubbard model goes over in the t-J modelrelating the parameters of the two models by t2/U = J/2. For µ = 0 it simplifies further tothe antiferromagnetic Heisenberg model relating the parameters by J = 2t2/U > 0.

As we are interested in antiferromagnetism, we will now describe how the Hubbard modelleads to the antiferromagnetic Heisenberg model and put U ≫ t and µ = 0. This situation canbe analyzed by a second order perturbation calculation in t/U , where the Coulomb part of theHubbard Hamiltonian from which we know the eigenstates, is the unperturbed Hamiltonianwhile the kinetic term is the perturbation. Since we have U ≫ t, doubly occupied sites becomeenergetically very unfavourable. At leading order (t = 0), i.e. neglecting the kinetic term, weget a ground state with one fermion per site. There are infinitely many such ground states fort = 0, as each spin can arbitrarily point up or down. When we take into account correctionsdue to hopping, this enormous degeneracy disappears. The first order correction is zero,because after one hop we do not return to a ground state. At second order, a fermion canfirst virtually hop to a nearest neighbour and then, one of them can hop back. Because hopsto nearest neighbours with parallel spin orientation are forbidden by the Pauli principle andbecause the corrections of second order are negative, this leads to antiparallel spin alignment.This perturbation calculation results in the antiferromagnetic Heisenberg Hamiltonian withJ related to the parameters of the Hubbard model by J = 2t2/U > 0 [43].

Therefore we have to work in the strong coupling limit and at half-filling to describe aHeisenberg antiferromagnet. As we also want to describe moving electrons and holes, it is notenough to take the Heisenberg model where the particles are fixed to the lattice, but we haveto use the Hubbard model. Looking at fig. 1.1, we see that an antiferromagnet does not getdestroyed immediately by doping. Hence, also at low doping we deal with an antiferromagnet.

3.5 Symmetries of the Hubbard Model

Since we want to describe an antiferromagnet on the honeycomb lattice with the Hubbardmodel, it should also have all symmetries of the underlying lattice which we figured out insection 2.2. In this section we show that the Hubbard model does so and that it has additionalsymmetries.

3.5.1 SU(2)s Spin Rotation Symmetry

The total SU(2)s spin operator (the subscript s stands for spin) is given by

~S =∑

x

~Sx =∑

x

c†x~σ

2cx. (3.8)

Later we need to know how the spinors transform under the various transformations as we willintroduce an operator which consists of them (see eq. (3.17)). Here it also serves to show thatthe discussed transformations are really symmetry transformations. To see how the spinorstransform, we need the unitary operator which implements the global spin rotation in theHilbert space of the theory

V = exp(

i~η · ~S)

. (3.9)

12 3. The underlying Microscopic Models and their Symmetries

Then the transformed spinors look like

c′x = V †cxV = exp

(

i~η · ~σ2

)

cx = gcx, with g = exp

(

i~η · ~σ2

)

∈ SU(2)s. (3.10)

One can see that the Hamiltonian is left invariant under spin rotation, i.e. [H, ~S] = 0, whichmeans that the total spin is conserved.

3.5.2 U(1)Q Fermion Number Symmetry

The U(1)Q fermion number operator is defined by

Q =∑

x

Qx =∑

x

(

c†xcx − 1)

=∑

x

(

c†x↑cx↑ + c†x↓cx↓ − 1)

, (3.11)

and counts the fermions relative to half-filling (due to the subtraction of the constant 1).Again we look at the unitary operator which implements the symmetry transformation in theHilbert space of the theory

W = exp (iωQ) , (3.12)

to see how the spinors transform

Qcx = W †cxW = exp (iω) cx, with exp (iω) ∈ U(1)Q. (3.13)

It is easy to see that the Hamiltonian is left invariant under this transformation, i.e. [H, Q] = 0.This means that the fermion number and with it the electric charge is conserved. Also inactual antiferromagnets this symmetry is unbroken until a superconducting phase is reached.

3.5.3 Discrete Symmetries Di, O, O′, and R

The spinors transform in the following ways

Di : Dicx = D†i cxDi = cx+ai ,

O : Ocx = O†cxO = cOx,

R : Rcx = R†cxR = cRx. (3.14)

For our purpose it is not important to know how the unitary operators look like that implementthe transformations in the Hilbert space. By redefining the sum over lattice points, one cansee that the Hubbard Hamiltonian is left invariant and the three transformations are indeedsymmetries. In non-relativistic theories orbital angular momentum and spin are separatelyconserved, the spin plays the role of an internal quantum number. This means that spatialrotations do not affect the spin and SU(2)s transformations can be applied independent ofO.

For later convenience it is useful to introduce a combination of the rotation with a specialSU(2)s transformation g = iσ2, which we call O′. It acts on the spinors as

O′

cx = O′†cxO′ = (iσ2)

Ocx = (iσ2) cOx. (3.15)

In the square lattice case we had a D′ instead of an O′. The Hubbard Model also has a time-reversal symmetry T . Since this is implemented by an antiunitary operator on the spinors,we will implement time-reversal later.

3.6 Manifestly SU(2)s ⊗ SU(2)Q Invariant Formulation of the Hubbard Model 13

3.5.4 Non-Abelian Extension of the U(1)Q Symmetry to SU(2)Q

Yang and Zhang noted that on bipartite lattices at half-filling the Hubbard model possessesan SU(2)Q symmetry, which is an extension of the U(1)Q symmetry discussed before [26,27].Away from µ = 0 the SU(2)Q symmetry is explicitly broken down to U(1)Q. Often it is calleda pseudospin symmetry and it is generated by the three operators

Q+ =∑

x

(−1)xc†x↑c†x↓, Q− =

∑

x

(−1)xcx↓cx↑,

Q3 =∑

x

1

2(c†x↑cx↑ + c†x↓cx↓ − 1) =

1

2Q. (3.16)

The factor (−1)x is defined to be 1 for x ∈ A and −1 for x ∈ B. In [2] it is discussed thatSU(2)Q is indeed a symmetry of the Hubbard model and there one can also see that next-to-nearest neighbour hopping breaks it. Explicitly it describes a particle-hole symmetry, whichmeans that the positive energy spectrum (for electrons) is the same as the negative one (forholes). Doping (µ 6= 0) induces a shift in this spectrum and destroys the symmetry. In actualmaterials this symmetry is not realized, but it helps to construct the effective theory and willbe broken down explicitly to U(1)Q in this process, such that it is not included in the effectivetheory.

3.6 Manifestly SU(2)s ⊗ SU(2)Q Invariant Formulation of the

Hubbard Model

As presented in eq. (3.1), the Hubbard Hamiltonian is not manifestly SU(2)Q invariant. Itturns out to be useful to introduce a matrix-valued fermion operator

Cx =

(

cx↑ (−1)x c†x↓cx↓ − (−1)x c†x↑

)

. (3.17)

A detailed derivation of this operator can be found in [2]. With its help it is possible to writethe Hubbard Hamiltonian in a manifestly SU(2)s ⊗ SU(2)Q invariant form. The unitaryoperator of the SU(2)Q transformation acting on the matrix-valued fermion operator reads

W = exp(

i~ω · ~Q)

, (3.18)

and the transformation yields

~QCx = W †CxW = CxΩT , with Ω = exp

(

i~ω · ~σ2

)

∈ SU(2)Q. (3.19)

Using the transformation rules of cx the remaining ones for Cx can be derived to be

SU(2)s : C ′x = gCx,

Di : DiCx = Cx+ai ,

O : OCx = COxσ3,

O′ : O′

Cx = (iσ2)COxσ3,

R : RCx = CRx. (3.20)

14 3. The underlying Microscopic Models and their Symmetries

When we make an SU(2)s as well as an SU(2)Q transformation, we get

~QC ′x = gCxΩ

T . (3.21)

Since SU(2)s acts from the left and SU(2)Q from the right, one can see that the two transfor-mations commute. This makes it possible to formulate the Hubbard Hamiltonian for µ = 0in the manifestly SU(2)s ⊗ SU(2)Q invariant form

H = − t

2

∑

〈xy〉Tr[

C†xCy + C†

yCx

]

+U

12

∑

x

Tr[

C†xCxC

†xCx

]

− µ

2

∑

x

Tr[

C†xCxσ3

]

. (3.22)

Chapter 4

Investigation of the Free Fermion

Model

In this chapter we take a look at the weak coupling limit of the Hubbard model at half filling.For U = 0, i.e. describing free fermions, the corresponding Hamiltonian can be diagonalizedanalytically. This yields the energy-momentum relation and we will see that the low-energyexcitations of such a system are free, relativistic, massless Dirac fermions. Also an analyticexpression for their velocity is derived. As there is no spontaneous breaking of a continuoussymmetry, in this case there are no Goldstone bosons. We then construct the leading terms ofthe corresponding effective Lagrangian for the Dirac fermions. This is possible although thereis no spontaneous breaking of a continuous symmetry as in BχPT. One only has to includeall relevant degrees of freedom in the effective theory which are present at low energies andhere these are only the Dirac fermions.

Graphene, a single sheet of graphite, i.e. carbon atoms arranged on a honeycomb lattice, isdescribed by the Hubbard model in the weak coupling limit at half filling. If some variants ofgraphene exist at stronger coupling, one eventually expects a phase transition at some criticalvalue of U , separating graphene’s unbroken phase from a strong coupling antiferromagneticphase, in which the SU(2)s spin symmetry is spontaneously broken to U(1)s. The Hubbardmodel for the square lattice case does not have such a phase transition as this describes anantiferromagnet for arbitrarily small values of U .

4.1 The Weak Coupling Limit of the Hubbard Model and how

it leads to Massless Dirac Fermions

In this section we investigate the weak coupling limit (U ≪ t) of the Hubbard model at half-filling (µ = 0). This will allow us to derive the energy-momentum relation in this limit. ForU ≪ t the interaction is only a very weak perturbation such that the fermions will behavealmost free. Thus in the following we treat the free case and put U = 0. By doing this, theHubbard Hamiltonian reduces to the kinetic part

H = Ht + Ht′ = −t∑

〈x,y〉

(

c†xcy + c†ycx)

− t′∑

〈〈x,y〉〉

(

c†xcy + c†ycx)

, (4.1)

where we also included a next-to-nearest neighbour hopping term proportional to t′ whichexplicitly breaks the SU(2)Q symmetry for t′ 6= 0. The symbol 〈〈x, y〉〉 indicates the sum

15

16 4. Investigation of the Free Fermion Model

over the next-to-nearest neighbours. Because we neglect the interaction part, we can di-agonalize the Hamiltonian by going to momentum space and derive the energy-momentumrelation. First we decompose the sum over the nearest and next-to-nearest neighbours intosums over the sublattices and appropriate vectors connecting the sites, such that we catcheach connection exactly once

H = − t∑

ji

∑

x∈A

(

cA†x cBx+ji + cB†x+ji

cAx

)

− t′∑

j′i

[

∑

x∈A

(

cA†x cAx+j′i+ cA†

x+j′icAx

)

+∑

x∈B

(

cB†x cBx+j′i

+ cB†x+j′i

cBx

) ]

, (4.2)

with A and B on the spinors indicating whether the fermions are created or annihilated onthe A or B sublattice. The vectors

j1 = (a, 0) , j2 =(

−12a,

√3

2 a)

, j3 =(

−12a,−

√3

2 a)

, (4.3)

connect the nearest neighbours and

j′1 =(

32a,

√3

2 a)

, j′2 =(

0,√

3a)

, j′3 =(

32a,−

√3

2 a)

, (4.4)

connect the next-to-nearest neighbours. In the following, one must take into account that theposition space is discrete because we work on a lattice. We use the relations

1

ABZ

∑

x

exp(

i(k − k′)x)

= δ(2)(k − k′),1

ABZ

∫

BZ

d2k exp(

ik(x− x′))

= δx,x′ , (4.5)

and for the spinors

cXx =1

ABZ

∫

BZ

d2k exp (ikx) cXk , X ∈ A,B, (4.6)

which we found by studying the square lattice case described in [43]. The components of the

spinors in momentum space cXk and cX†k still obey the standard anticommutation relations.

ABZ is the area of the first Brillouin zone (BZ) which is given in eq. (2.5). By replacing thecreation and annihilation operators in eq. (4.2) by the ones in momentum space and usingthe other relations we get

H = − 1

ABZ

∫

BZ

d2k(

cA†k , cB†k

)

H(k)

(

cAkcBk

)

, (4.7)

with

H(k) =

(

t′f(k) tg(k)tg∗(k) t′f(k)

)

, (4.8)

and

f(k) = 4 cos

(

3

2k1a

)

cos

(√3

2k2a

)

+ 2cos(√

3k2a)

, (4.9)

g(k) = cos (k1a) + i sin (k1a) + 2 cos

(√3

2k2a

)

[

cos

(

1

2k1a

)

− i sin

(

1

2k1a

)]

. (4.10)

4.1 The Weak Coupling Limit of the Hubbard Model and how it leads to Massless Dirac Fermions 17

When we diagonalize the Hamiltonian of eq. (4.8), we get the energy-momentum relation forfree fermions

E±(k) = t′f(k) ± t|g(k)| = t′f(k) ± t√

3 + f(k)

= t′[

4 cos

(

3

2k1a

)

cos

(√3

2k2a

)

+ 2cos(√

3k2a)

]

± t

√

√

√

√3 + 4 cos

(

3

2k1a

)

cos

(√3

2k2a

)

+ 2cos(√

3k2a)

. (4.11)

The upper solution corresponds to the conduction and the lower to the valence band. Fig. 4.1shows the energy-momentum relation for t = 1 and t′ = 0. In this case the SU(2)Q symmetryis not broken and the two bands are symmetric with respect to the plane with zero energy.At zero temperature, since we are at half-filling, all states with energies in the valence bandare covered, while all in the conduction band are empty. The two bands touch each otherat the corners of the first Brillouin zone at the height of zero energy, which explains thesemimetallic behaviour of graphene. These six points, known as Dirac points, represent theFermi surface at EF = 0 for t′ = 0. When only t varies, the energy scale changes but theshape of the bands remains the same. From eq. (4.11) it is clear that for t = 0 the bandsdegenerate for all values of t′. If t′ is nonzero, the SU(2)Q symmetry is explicitly broken, the

-20

2

k1

-20

2k2

-2

0

2

EHkL

Figure 4.1: Energy-momentum relation of free fermions with t = 1 and t′ = 0.

two bands are not symmetric anymore and the energy value where the two bands touch, isshifted. Two examples with t′ 6= 0 are shown in fig. 4.2. For some values of t and t′, someenergy region is covered by both bands, which implies that the Fermi surface is not at theDirac points anymore. We showed that the two bands touch at the six Dirac points for allvalues of t and t′.

Now we expand the energy-momentum relation of eq. (4.11) at each Dirac point in aTaylor series up to first order. This yields

E±(p) = ±3ta

2

√

p21 + p2

2 − 3t′ + O(p2), (4.12)

with the momenta p1 and p2 now measured relative to the corresponding Dirac point. Becausethis is a cone and because this is exactly the relativistic energy-momentum relation of a

18 4. Investigation of the Free Fermion Model

-20

2

k1

-20

2k2

0

2

4

EHkL

(a) t = 1, t′ = 0.2

-2

0

2

k1

-20

2k2

-10

-5

0

5

EHkL

(b) t = 1, t′ = −1.5

Figure 4.2: Energy-momentum relation of free fermions with different values of t and t′.

massless, free particle with velocity c (up to an irrelevant shift −3t′)

E±(p) = ±c|p| = ±c√

p21 + p2

2, (4.13)

they are called Dirac cones. (This is also the reason why the points are called Dirac points.)Hence, when the Dirac points are the Fermi surface, at low energies the microscopic theorypredicts massless, free, relativistic fermions having a velocity

vF =3ta

2. (4.14)

It is astonishing that the velocity does not depend on the momentum of the particle. Its valueis about 106 m/s [35]. These strange massless Dirac fermions are a product of the geometryof the honeycomb lattice, which is incorporated in the energy-momentum relation via thevectors in eq. (4.3).

Free relativistic fermions with zero mass in two space dimensions are described by theDirac Hamiltonian

HD(p) = αpc = (−σ2p1 + σ1p2) c, (4.15)

with αi = γ0γi, where we have chosen the gamma matrices to be

γ0 = σ3, γ1 = iσ1, γ2 = iσ2, (4.16)

satisfying

γµ, γν = 2gµν12×2, µ, ν ∈ 0, 1, 2, with gµν =

1 0 00 −1 00 0 −1

. (4.17)

Here c denotes the velocity of the particle. We have shown in addition to the above derivationthat expanding eq. (4.8) with t′ = 0 up to first order yields exactly the above Dirac Hamilto-nian. The set of gamma matrices may differ in each corner by an unitary transformation.

4.2 Effective Field Theory for Dirac Fermions 19

4.2 Effective Field Theory for Dirac Fermions

Now we will construct a low-energy effective field theory for the Dirac fermions for t′ = 0. Forsimplicity we ignore spin because in the free case it just represents a trivial irrelevant label.As we do not consider spin, we do not have the SU(2)S symmetry. In addition, the generatorsQ+ and Q− of the SU(2)Q symmetry are not defined and instead we have the U(1)Q and anadditional Z(2) symmetry, which represents a discrete particle-hole symmetry, a remnant ofthe SU(2)Q symmetry. It will be interesting to see that mass terms are forbidden because ofsymmetries alone.

With the following procedure we change from a Hamiltonian to a Lagrangian formulationused in a Euclidean path integral. We will deal with fermion fields represented by Grassmannnumbers defined in the space-time continuum and no longer with microscopic operators de-fined on the lattice. From now on xmeans a Euclidean space-time point x = (x1, x2, t). Beforewe can formulate the effective theory, we have to identify the correct fermion fields and derivetheir transformation behaviour under the various symmetry transformations.

4.2.1 Fermion Fields

With the help of the energy-momentum relation, we can conclude that at half-filling thelowest energy states are in the circular shaped neighbourhoods of the corners of the firstBrillouin zone in momentum space, which are called pockets. Hence, the effective theory hasto describe fermions living in these pockets. As already mentioned in section 2.1, there areonly two inequivalent corners of the Brillouin zone. The transformation rules of the fermionfields turn out to be simple, when we chose the corners

k2 = kα =(

0, 4π3√

3a

)

, and k5 = kβ =(

0,− 4π3√

3a

)

, (4.18)

which we call kα and kβ in the following. Their circular shaped neighbourhoods are denotedas the α and β pocket. For completeness we also have to include the origin Γ. In orderto address these points in momentum space, we introduce further sublattices in positionspace which are superimposed with the already introduced A and B sublattices, see fig.4.3. A fermion at space-time point x is represented by ψXi(x), where Xi denotes one of thesublattices introduced above. The discrete lattice in position space and the Brillouin zoneare connected by a Fourier transformation. With these new fields it is possible to definefermion fields with a momentum index by a discrete Fourier transformation connecting thethree points of the corresponding sublattice in position space with the three desired pointsin momentum space. (As we are not interested to describe fermions at Γ, we do not considerthe corresponding field.)

ψA,f (x) =1√3

3∑

j=1

exp(

−ikfvj)

ψAj (x),

ψB,f (x) =1√3

3∑

j=1

exp(

−ikfwj)

ψBj (x), (4.19)

20 4. Investigation of the Free Fermion Model

x1

x2

x1

x2

B3 B3

B1 B1 B1

B2 B2 B2

B3 B3 B3

B1 B1

B1

B2 B2 B2

B3 B3 B3

B1 B1 B1

B2 B2 B2

B3

A1

A2 A2 A2

A3 A3 A3

A1 A1 A1

A2 A2 A2

A3

A3 A3

A1 A1 A1

A2 A2 A2

A3 A3 A3

A1 A1

Figure 4.3: Further sublattices and the corresponding primitive lattice vectors.

with the ”flavour” f ∈ α, β and the vectors vj and wj (see also fig. 4.4)

v1 =(

−12a,−

√3

2 a)

, v2 = (a, 0) , v3 =(

−12a,

√3

2 a)

,

w1 =(

12a,−

√3

2 a)

, w2 = (−a, 0) , w3 =(

12a,

√3

2 a)

. (4.20)

w3v3

w2

v1 w1

v2

Figure 4.4: Vectors from eq. (4.20).

These fields now describe fermions living in the α and β pocket and the indices A and Brepresent an additional quantum number. To summarize, we have the fields

ψA,α(x) =1√3

(

exp(i2π3 )ψA1(x) + ψA2(x) + exp(−i2π3 )ψA3(x))

,

ψA,β(x) =1√3

(

exp(−i2π3 )ψA1(x) + ψA2(x) + exp(i2π3 )ψA3(x))

,

ψB,α(x) =1√3

(

exp(i2π3 )ψB1(x) + ψB2(x) + exp(−i2π3 )ψB3(x))

,

ψB,β(x) =1√3

(

exp(−i2π3 )ψB1(x) + ψB2(x) + exp(i2π3 )ψB3(x))

. (4.21)

4.2 Effective Field Theory for Dirac Fermions 21

The corresponding conjugate fields ψX,f†(x) are obtained by a dagger operation. However,one must keep in mind that the conjugate Grassmann fields are independent (in contrast tothe microscopic operators in Hilbert space).

Next we derive the transformation properties. Due to the different nature of the micro-scopic theory and the effective theory, it is not possible to derive a connection between thetwo formulations in a rigorous way. Our connection between the two is to demand that theGrassmann fields inherit the transformation properties of the microscopic operators, which wehave already worked out in section 3.5. Hence, the fields on the right hand side of eq. (4.21)transform exactly like the corresponding microscopic operators cXix (without spin). How thesublattices Xi change under the geometric symmetries can be seen by considering how thesublattices map onto each other. We already mentioned the Z(2) symmetry, which acts onthe microscopic operators as

Z(2)cAix = cAi†x , Z(2)cAi†x = cAix ,

Z(2)cBix = −cBi†x , Z(2)cBi†x = −cBix . (4.22)

The effective theory is valid only at low energies. As short distances correspond to highmomenta, it does not care about distances of the order of the lattice spacing and we thusno longer distinguish between the points x and x+ ai under shift transformations in theframework of the effective theory. Time-reversal is realized on the Grassmann fields in theusual manner. Thus we get the following transformation rules of the fermion fields

Z(2) : Z(2)ψA,f (x) = ψA,f′†(x),

Z(2)ψB,f (x) = −ψB,f ′†(x),U(1)Q : QψX,f (x) = exp(iω)ψX,f (x),

Di : DiψX,f (x) = exp(ikfai)ψX,f (x),

O : OψA,α(x) = exp(−i2π3 )ψB,β(Ox),

OψA,β(x) = exp(i2π3 )ψB,α(Ox),OψB,α(x) = exp(i2π3 )ψA,β(Ox),OψB,β(x) = exp(−i2π3 )ψA,α(Ox),

R : RψX,f (x) = ψX,f′

(Rx),

T : TψX,f (x) = −ψX,f ′†(Tx),TψX,f†(x) = ψX,f

′

(Tx), (4.23)

with f ′ being the flavour other than f and X ∈ A,B. Again, apart from time-reversal, onegets the conjugate fields by a dagger operation. O, R and T act on a space-time point as

Ox =(

12x1 −

√3

2 x2,√

32 x1 + 1

2x2, t)

,

Rx = (x1,−x2, t),

Tx = (x1, x2,−t). (4.24)

4.2.2 Effective Lagrangian

Since the appropriate fermion fields are found and their transformation rules are derived, weare now ready to construct the leading terms of the systematic low-energy effective Lagrangian

22 4. Investigation of the Free Fermion Model

for the Dirac fermions. More concretely, this means that we systematically investigate allpossible terms consisting of the fermion fields with momentum index, their conjugate partners,and derivatives. As mentioned in the introduction, we only need the lowest order of thederivative expansion to describe the low-energy physics. Since the fermions to be describedhave a relativistic energy-momentum relation (E ∝ p) at low energies, temporal and spatialderivatives count the same. This will be different when we construct the effective theoryfor magnons and holes in the antiferromagnet. Then all terms which are invariant under allsymmetry transformations are endowed with a low-energy constant and form together thedesired Lagrangian which has all symmetries of the underlying Hubbard Hamiltonian. Wecategorize the terms according to the number of derivatives and the number of fields nψ theyconsist of. The number of fields and derivatives does not change under the transformations.Thus we can investigate each class of terms separately. Because of the U(1)Q symmetry,only terms with the same number of undaggered and daggered fields are possible, otherwisethere would remain a general phase exp(±iω) when applying the transformation. One musttake care that the remaining terms are independent. For example, by partial integration (inthe action), from ∂tψ

X,f†ψX,f one gets −ψX,f†∂tψX,f and a boundary term which is zeroas the fields go to zero at infinity. Therefore one may take only one of the two terms. Inthe framework of free Dirac fermions (still for µ = 0 and t′ = 0) we limit ourselves to theLagrangian with two fermion fields and up to one derivative

Lfree2 =∑

f=α,βX=A,B

[

ψX,f†∂tψX,f + vF

(

σXψX,f†∂1ψ

X′,f + iσfψX,f†∂2ψ

X′,f)]

, (4.25)

with X ′ being the sublattice index different from X and

σX =

1 for X = A

−1 for X = B, and σf =

1 for f = α

−1 for f = β. (4.26)

This Lagrangian describes the kinetic energy up to first order of a massless, free Dirac fermionliving at the α or β pocket. Here the low-energy constant of the term with the temporalderivative is normalized to one and the only low-energy constant is the fermion velocity vF .As we see, the above Lagrangian contains no mass terms as predicted by the underlyingHubbard model, which shows the strength of the effective theory which is just based on asymmetry analysis.

In semimetallic graphene the fermions are, of course, not completely free but coupledweakly. Hence, to describe graphene, in addition to including the spin, one would also haveto construct the leading terms with more than two fermion fields which describe contactinteractions, like the 4-Fermi terms. But terms with identical fermion fields vanish because ofthe Pauli principle and since we have eight different fields, terms with more than eight fieldsand without derivatives do not exist.

Massless, free fermions are described by the Dirac Lagrangian without mass terms (hereusing a (2 + 1)-dimensional Euclidean metric)

LD = cΨγµ∂µΨ, with Ψ = Ψ†γ3. (4.27)

c denotes again the velocity of the particle and the gamma matrices are given by

γj = iσj . (4.28)

4.2 Effective Field Theory for Dirac Fermions 23

Our Lagrangian can be brought in this form in the following way. First one puts the fieldsinto the spinors

Ψα(x) =

(

ψA,α(x)ψB,α(x)

)

, Ψβ(x) =

(

ψA,β(x)ψB,β(x)

)

. (4.29)

When we then multiply both Lagrangians with 1/vF and 1/c, respectively, there exists aunitary transformation of the gamma matrices such that our Lagrangian assumes the formof the Dirac Lagrangian under the condition c = −vF . Using eq. (4.14), we see that a signchange of vF implies a sign change of the hopping amplitude t. This interchanges particlesand holes (see eq. (3.1)), which does not change the physics. Therefore the Lagrangian canbe written in the form of the Dirac Lagrangian with one term for the α and one term for theβ pocket

Lfree2 = vF

(

Ψαγµ∂µΨα + Ψβγµ∂µΨ

β)

. (4.30)

Here a temporal derivative also includes a factor 1/vF .

Chapter 5

Magnons

Now we return to the strong coupling limit, i.e. to antiferromagnets. A characteristic featureof quantum antiferromagnets is the spontaneous breaking of the SU(2)s symmetry to thesubgroup U(1)s, which gives rise to two massless Goldstone bosons, the magnons. Sincethey are, apart from the holes, the only relevant degrees of freedom at low-energies, theirlow-energy physics can be described by an effective field theory analogous to BχPT with orwithout holes.

This and the next chapter include all additional preparations for the construction ofthe effective theory for magnons and holes in an antiferromagnet on the honeycomb lattice,which will be developed in chapter 7. In this chapter we first describe the spontaneoussymmetry breaking. Then the leading terms of the Euclidean action for the pure magnonsector (corresponding to the pure pion sector of QCD) are derived. Finally, as in BχPT, weconstruct a non-linear realization of the spontaneously broken SU(2)s symmetry, which willbe used to couple the magnon fields to the hole fields later. By doing all this, we develop stepby step the adequate magnon fields and derive their transformation rules under the varioussymmetries of the Hubbard model.

5.1 Spontaneous Symmetry Breaking

First we take a look at the interesting phenomenon of spontaneous symmetry breaking. Wehave seen in section 3.4, that an antiferromagnet is already described by the Heisenberg model.For the ground state of the antiferromagnetic Heisenberg Hamiltonian on the honeycomblattice we would naively expect all spins on the A sublattice to point up and all spins on theB sublattice to point down or vice versa. This is the classical antiferromagnetic Neel state,which can be written as

|N〉 =∏

x∈Ac†

x↑∏

x∈Bc†

x↓ |0〉 . (5.1)

It motivates the definition of the staggered magnetization vector

~Ms =∑

x

(−1)x ~Sx, (5.2)

which is an order parameter. Note that in the Neel state the ordinary uniform magnetizationis zero. We already defined the factor (−1)x to be 1 for x ∈ A and −1 for x ∈ B. When definedthe other way round, the staggered magnetization would point in the opposite direction, but

25

26 5. Magnons

the physics remained the same. The existence of a non-vanishing order parameter indicatesa spontaneous symmetry breaking. Indeed, the Heisenberg Hamiltonian has an SU(2)s spinsymmetry, which can be seen by calculating

[H, ~S] = 0, (5.3)

where ~S is the operator of the total spin

~S =∑

x

~Sx. (5.4)

Instead of this SU(2)s symmetry, the classical Neel state is invariant only under U(1)s spinrotations around the spontaneously selected direction of the staggered magnetization.

Taking a closer look, one can see that the classical Neel state is not an eigenstate of theHeisenberg Hamiltonian. It can also be shown that the ground state is an SU(2)s singlet [41],which is not true for the classical Neel state. Therefore it can not be the ground state. Asalready mentioned, the true ground state is not known analytically, but it is known that thereare also spin fluctuations in addition to the strict antiparallel spin alignment.

Because we do not know the ground state analytically, it is not obvious that there is a spon-taneous symmetry breaking. It has been shown by numerical simulations that the antiferro-magnetic Heisenberg model indeed shows a spontaneous symmetry breaking SU(2)s → U(1)s,where the staggered magnetization is the appropriate order parameter. In BχPT there is alsosuch a spontaneous breaking of a continuous symmetry and it is possible to describe theHubbard model at low energies with an effective theory analogous to BχPT.

5.2 Effective Theory for Magnons

As just discussed, in quantum antiferromagnets there occurs a spontaneous symmetry break-ing from the group G = SU(2)s to the subgroup H = U(1)s. For such a case Goldstone’stheorem [13] predicts dim(G)− dim(H) = 3− 1 = 2 massless bosons, here two magnons, alsocalled antiferromagnetic spin waves. They can be interpreted as two linearly independentexcitation directions of spin waves. The Goldstone bosons are described by fields in the cosetspace G/H. In our case, the magnons are described by a classical unit-vector field ~e(x), livingin the coset space

G/H = SU(2)s/U(1)s ≡ S2, (5.5)

~e(x) =(

e1(x), e2(x), e3(x))

, ~e(x)2 = 1, (5.6)

with x again being a Euclidean space-time point. One can view the ~e(x) field as represent-ing the local staggered magnetization. As already mentioned, the effective theory does notcare about distances of the order of a lattice spacing, as such distances correspond to highmomenta. For the same reason the ~e(x) field represents an average over a lot of spins. Dueto the different nature of the ~e(x) field and the microscopic staggered magnetization ~Ms, therelation between them can not be derived in a rigorous way.

In order to derive a magnon action, we must know how the ~e(x) field transforms underthe various symmetries of the Hubbard model. Since it corresponds to the staggered magne-tization of the microscopic model, for the geometric transformations one only has to checkwhether the sublattices are interchanged or not. An SU(2)s spin rotation is realized on ~e(x)by a R ∈ SO(3)s transformation, where the SU(2)s transformation defines the corresponding

5.2 Effective Theory for Magnons 27

one in SO(3)s. How time-reversal is realized on ~e(x), can be derived as follows. The ~e(x) fieldcorresponds to the staggered magnetization and this to spin operators. Spin is a form of an-gular momentum ~L = ~r × ~p and under time-reversal the momentum changes sign. Thereforeone concludes that the ~e(x) field changes sign under time-reversal.

Because the SU(2)Q and U(1)Q symmetries act non-trivially only on fermions, not onbosons, the ~e(x) field and the following magnon field representations are invariant underthese transformations. To summarize, we have the following transformation properties

SU(2)S : ~e(x)′ = R~e(x),Di : Di~e(x) = ~e(x),

O : O~e(x) = −~e(Ox),R : R~e(x) = ~e(Rx),

T : T~e(x) = −~e(Tx), (5.7)

with Ox, Rx, and Tx as in eq. (4.24). As already mentioned, spatial and spin rotations donot affect each other. Especially the spin rotation does not affect the space-time point x. Insection 2.2 we pointed out that the sublattices change under rotation, thus we get a minussign, and they don’t change under a shift. This is different in the square lattice case.

Now we can construct the leading terms of the Euclidean action for the pure magnonsector as we did for the Dirac fermions. They describe the low-energy physics of an undopedantiferromagnet. Antiferromagnetic magnons have a relativistic energy-momentum relation(E ∝ p) at low energies. Therefore spatial and temporal derivatives count the same. Termslike ~e(x) · ~e(x) are constant and thus irrelevant, while terms like ~e(x) · ∂µ~e(x) are zero. Wefind exactly the same action as in the square lattice case [24,28–30]

S[~e ] =

∫

d2x dt L0 =

∫

d2x dtρs2

(

∂i~e · ∂i~e+1

c2∂t~e · ∂t~e

)

, (5.8)

where the index i ∈ 1, 2 (summation) stands for the two spatial directions. The Euclideantime direction is compactified to a circle S1 of circumference β = 1/T , where T is the temper-ature of the system. The spin stiffness ρs and the spin wave velocity c are material-dependentlow-energy constants. For energy values small compared to ρs, the low-energy expansion isvalid. The values of the low-energy constants can not be derived in the framework of theeffective theory. They have to be determined by either experiments or numerical simulations.The two constants ρs and c have been determined very precisely by fitting Monte Carlo datato predictions of the effective theory in [25] for the square and in [10] for the honeycomblattice and are given in tab. 5.1. As mentioned in section 2.1, the coordination number of thehoneycomb lattice is three, while for the square lattice it is four, which explains the smallervalues for the honeycomb lattice as the spins are more weakly coupled.

In [32–34] it is shown, that there is no spontaneous symmetry breaking for systems in twodimensions. Together with time, we are working in 2 + 1 dimensions, where the extent of thetime dimension is inversely proportional to the temperature T . Thus, at zero temperature weare dealing with three dimensions, where spontaneous symmetry breaking may occur. At non-zero temperatures, the extent of the time dimension is finite and the magnons pick up a masswhich is exponentially small in the inverse temperature [30]. The effective theory remainsvalid as long as the described particles are the lightest and remain the relevant degrees offreedom in the low-energy spectrum.

28 5. Magnons

Lattice ρs c

Square 0.1808(4)J 1.6585(10)Ja

Honeycomb 0.102(2)J 1.297(16)Ja

Table 5.1: Numerical values for the low-energy constants ρs and c. J is the exchange couplingconstant of the Heisenberg model and a the corresponding lattice spacing.

Instead of working with the vector field representation ~e(x) of the magnon field, it willturn out to be more convenient to use a 2 × 2 matrix representation P (x) defined by

P (x) =1

2

(1+ ~e(x) · ~σ)

=1

2

(

1 + e3(x) e1(x) − ie2(x)e1(x) + ie2(x) 1 − e3(x)

)

, (5.9)

with P (x) ∈ CP (1) ∼= S2 having the properties

P (x)† = P (x), TrP (x) = 1, P (x)2 = P (x), (5.10)

i.e. it is a Hermitian projection matrix. On the matrix representation P (x), the SU(2)stransformation is again realized by an element g ∈ SU(2)s as shown below. The behaviourof P (x) under the other transformations can be derived from the transformation rules ofthe ~e(x) field. For later convenience we also introduce a O′ and a T ′ transformation whichare the ordinary rotation and time-reversal transformations combined with a special SU(2)stransformation g = iσ2. Then we obtain the following transformation behaviour

SU(2)S : P (x)′ = gP (x)g†,

Di : DiP (x) = P (x),

O : OP (x) = 1− P (Ox),

O′ : O′

P (x) = P (Ox)∗,

R : RP (x) = P (Rx),

T : TP (x) = 1− P (Tx) = OP (O−1Tx),

T ′ : T ′

P (x) = P (Tx)∗ = O′

P (O−1Tx). (5.11)

The transformed objects have the same properties as P (x). Written in terms of P (x), themagnon action reads

S[P ] =

∫

d2x dt ρsTr

[

∂iP∂iP +1

c2∂tP∂tP

]

. (5.12)

5.3 Non-linear Realization of the SU(2)s Symmetry

Since SU(2)s is spontaneously broken, global SU(2)s transformations on the hole fields willbe realized by a local, non-linear symmetry transformation in the unbroken subgroup U(1)s.This concept, where a spontaneously broken symmetry group is represented by a non-linearrealization in a continuous subgroup, is used in BχPT and can be generalized to arbitrarycompact, connected, semi-simple Lie groups [15,16]. With this procedure, the magnon fields

5.3 Non-linear Realization of the SU(2)s Symmetry 29

can be coupled to the hole fields. The non-linear realization is constructed from the globaltransformation g ∈ SU(2)s and P (x). In a first step, we uniquely represent the magnon fieldby a local SU(2)s field u(x) which diagonalizes P (x) by a unitary transformation

u(x)P (x)u(x)† =1

2(1+ σ3) =

(

1 00 0

)

, u11(x) ≥ 0. (5.13)

To obtain a well-defined diagonalizing field u(x), we demand that u11 is real and non-negative.Otherwise it would be defined only up to a U(1)s phase. A detailed derivation of u(x) can befound in [2]. One gets

u(x) =1

√

2(1 + e3(x))

(

1 + e3(x) e1(x) − ie2(x)−e1(x) − ie2(x) 1 + e3(x)

)

=

(

cos(θ(x)2 ) sin(θ(x)2 ) exp(−iϕ(x))

− sin(θ(x)2 ) exp(iϕ(x)) cos(θ(x)2 )

)

. (5.14)

For the second equality spherical coordinates are used

~e(x) =(

sin θ(x) cosϕ(x), sin θ(x) sinϕ(x), cos θ(x))

. (5.15)

As the diagonalizing field rotates an arbitrary magnon field configuration P (x) at each space-time point in the constant configuration of eq. (5.13), which corresponds to ~e(x) = (0, 0, 1),it contains the same information as P (x) or ~e(x).

With u(x) we have an SU(2)s representation of the magnon field. Now we are lookingfor the local non-linear symmetry transformation h(x) in the unbroken subgroup U(1)s. Itresults from the behaviour of u(x) under global SU(2)s transformations. We demand

u(x)′P (x)′u(x)′† = u(x)P (x)u(x)† =1

2(1+ σ3). (5.16)

Since P (x) transforms under an SU(2)s transformation as gP (x)g†, one might think that thetransformed u(x) takes the form u(x)′ = u(x)g†. However, then the uniqueness of u(x) afterthe transformation is again not guaranteed as u11 may pick up a U(1)s phase. In order tomake it unique also after the transformation, we must introduce a local non-linear symmetrytransformation h(x), which makes u11 real and non-negative. It has the form

h(x) = exp(

iα(x)σ3

)

=

(

exp(iα(x)) 00 exp(−iα(x))

)

∈ U(1)s. (5.17)

Then the diagonalizing field u(x) transforms under SU(2)s as

u(x)′ = h(x)u(x)g†, u11(x)′ ≥ 0, (5.18)

which implicitly defines h(x). The x-dependence of h(x) arises because it depends on P (x).However this does not mean that the SU(2)s transformation has become truly local. Withnon-linearity is meant, that h(x) depends on the magnon field. In the special case that g is inthe unbroken subgroup (g = diag

(

exp(iβ), exp(−iβ))

∈ U(1)s), h(x) reduces to h(x) = h = gi.e. it becomes global and linearly realized.

30 5. Magnons

It remains to show that the SU(2)s group structure is properly inherited by the non-linearU(1)s realization. This is the case when a composite SU(2)s transformation g = g2g1 leadsto a composite transformation h(x) = h2(x)h1(x). The first transformation g1 defines h1

P (x)′ = g1P (x)g†1, u(x)′ = h1(x)u(x)g†1. (5.19)

Then the second transformation g2 defines h2

P (x)′′ = g2P (x)′g†2 = g2g1P (x)(g2g1)† = gP (x)g†,

u(x)′′ = h2(x)u(x)′g†2 = h2(x)h1(x)u(x)(g2g1)

† = h(x)u(x)g†. (5.20)

Indeed h(x) = h2(x)h1(x) and thus the group structure is properly inherited.Now we know how the SU(2)s transformation is realized on the diagonalizing field u(x).

Under the other transformations one finds

Di : Diu(x) = u(x),

O : Ou(x) = τ(Ox)u(Ox),

O′ : O′

u(x) = u(Ox)∗,

R : Ru(x) = u(Rx),

T : Tu(x) = τ(Tx)u(Tx) = Ou(O−1Tx),

T ′ : T ′

u(x) = u(Tx)∗ = O′

u(O−1Tx), (5.21)

with

τ(x) =1

√

e1(x)2 + e2(x)2

(

0 −e1(x) + ie2(x)e1(x) + ie2(x) 0

)

=

(

0 − exp(−iϕ(x))exp(iϕ(x)) 0

)

∈ U(1). (5.22)

One can see that the transformations, under which the ~e(x) field changes sign (O and T ),are also spontaneously broken as they are realized in a non-linear manner (τ(x) depends onthe magnon field). The combined transformations O′ and T ′ are realized in a linear manner(independent of the magnon field) and are thus not spontaneously broken. We introducedthem because they are easier to handle, as their transformation behaviour is simpler. They areused instead of O and T to facilitate the construction of terms in the effective Lagrangian. Inthe square lattice case, instead of the rotation, the shift symmetry is spontaneously broken.As already mentioned, this is because on the square lattice the sublattices A and B areinterchanged by a shift, while on the honeycomb they are interchanged under rotation.

The above transformation rules can be derived from the transformations of the ~e(x)field. For the combined transformations one first applies O or T , respectively, and thenthe special SU(2)s transformation, where h(x) takes the form h(Ox) = (iσ2)τ(Ox)

† orh(Tx) = (iσ2)τ(Tx)

†, respectively.

5.4 Composite Magnon Field vµ(x)

All still in analogy to BχPT, we introduce the anti-Hermitian composite magnon field vµ(x)

vµ(x) = u(x)∂µu(x)†. (5.23)

5.4 Composite Magnon Field vµ(x) 31

As it is constructed from the diagonalizing field u(x), it is just another magnon field repre-sentation. One finds the transformation rules of vµ(x) using those of u(x)

SU(2)s : vµ(x)′ = h(x)[vµ(x) + ∂µ]h(x)

†,

Di : Divµ(x) = vµ(x),

O : Ov1(x) = τ(Ox)12

(

v1(Ox) + ∂1 +√

3v2(Ox) +√

3∂2

)

τ(Ox)†,Ov2(x) = τ(Ox)1

2

(

−√

3v1(Ox) −√

3∂1 + v2(Ox) + ∂2

)

τ(Ox)†,Ovt(x) = τ(Ox)(vt(Ox) + ∂t)τ(Ox)

†,

O′ : O′

v1(x) = 12

(

v1(Ox)∗ +

√3v2(Ox)

∗),O′

v2(x) = 12

(

−√

3v1(Ox)∗ + v2(Ox)

∗),O′

vt(x) = vt(Ox)∗,

R : Rv1(x) = v1(Rx),Rv2(x) = −v2(Rx),Rvt(x) = vt(Rx),

T : T vi(x) = τ(Tx)(vi(Tx) + ∂i)τ(Tx)†,

T vt(x) = −τ(Tx)(vt(Tx) + ∂t)τ(Tx)†,

T ′ : T ′

vi(x) = vi(Tx)∗,

T ′

vt(x) = −vt(Tx)∗. (5.24)

Since the composite magnon field is anti-Hermitian and traceless, we can decompose it intothe Pauli matrices

vµ(x) = ivaµ(x)σa = i

(

v3µ(x) v+

µ (x)

v−µ (x) −v3µ(x)

)

, a ∈ 1, 2, 3, vaµ(x) ∈ R, (5.25)

where the factor i makes vµ(x) anti-Hermitian. The components vaµ(x) are given by

vaµ(x) =1

2iTr [vµ(x)σa] . (5.26)

They do not represent independent degrees of freedom as they are all built from the magnonfield. For convenience we also define the fields

v±µ (x) = v1µ(x) ∓ iv2

µ(x). (5.27)

The v3µ(x) and the v±µ (x) fields will finally be used. Let us first investigate their behaviour

under SU(2)s transformations. With the eqs. (5.24), (5.25), and (5.17) (where α(x) stemsfrom) one finds

v3µ(x)

′ = v3µ(x) − ∂µα(x),

v±µ (x)′ = exp(±2iα(x))v±µ (x). (5.28)

The transformation rule for v3µ(x) looks like the one of an Abelian U(1)s gauge field, while the

v±µ (x) show the behaviour of vector fields charged under U(1)s. Although the spin rotationsymmetry disguises itself as a gauge-like symmetry, one should stress that the spin symmetry

32 5. Magnons

is not really gauged. In fact, the gauge field nature of v3µ(x) results from the non-linear U(1)s

realization of the global SU(2)s symmetry.

Under the other transformations using eqs. (5.24) and (5.25) one finds the following be-haviour of the v3

µ(x) and v±µ (x) fields (ϕ(x) stems from the diagonalizing field u(x))

Di : Div3µ(x) = v3

µ(x),

O : Ov31(x) = 1

2

(

− v31(Ox) + ∂1ϕ(Ox) −

√3v3

2(Ox) +√

3∂2ϕ(Ox))

,

Ov32(x) = 1

2

(√

3v31(Ox) −

√3∂1ϕ(Ox) − v3

2(Ox) + ∂2ϕ(Ox))

,Ov3

t (x) = −v3t (Ox) + ∂tϕ(Ox),

O′ : O′

v31(x) = −1

2

(

v31(Ox) +

√3v3

2(Ox))

,

O′

v32(x) = 1

2

(√

3v31(Ox) − v3

2(Ox))

,

O′

v3t (x) = −v3

t (Ox),

R : Rv31(x) = v3

1(Rx),Rv3

2(x) = −v32(Rx),

Rv3t (x) = v3

t (Rx),

T : T v3i (x) = −v3

i (Tx) + ∂iϕ(Tx),T v3

t (x) = v3t (Tx) − ∂tϕ(Tx),

T ′ : T ′

v3i (x) = −v3

i (Tx),

T ′

v3t (x) = v3

t (Tx), (5.29)

Di : Div±µ (x) = v±µ (x),

O : Ov±1 (x) = − exp(∓2iϕ(Ox))12

(

v∓1 (Ox) +√

3v∓2 (Ox))

,

Ov±2 (x) = exp(∓2iϕ(Ox))12

(√

3v∓1 (Ox) − v∓2 (Ox))

,Ov±t (x) = − exp(∓2iϕ(Ox))v∓t (x),

O′ : O′

v±1 (x) = −12

(

v∓1 (Ox) +√

3v∓2 (Ox))

,

O′

v±2 (x) = 12

(√

3v∓1 (Ox) − v∓2 (Ox))

,

O′

v±t (x) = −v∓t (Ox),

R : Rv±1 (x) = v±1 (Rx),Rv±2 (x) = −v±2 (Rx),Rv±t (x) = v±t (Rx),

T : T v±i (x) = − exp(∓2iϕ(Tx))v∓i (Tx),T v±t (x) = exp(∓2iϕ(Tx))v∓t (Tx),

T ′ : T ′

v±i (x) = −v∓i (Tx),

T ′

v±t (x) = v∓t (Tx). (5.30)

It is shown in [2], how the magnon action can be formulated in terms of the composite magnon

5.4 Composite Magnon Field vµ(x) 33

field which then reads

S[v±µ ] =

∫

d2x dt 2ρs

(

v+i v

−i +

1

c2v+t v

−t

)

. (5.31)

As the fields contain a derivative, the terms in the action are kinetic terms and not massterms.

Chapter 6

From Microscopic Operators to

Effective Fields for Holes

Since we also want to describe doped holes, we have to derive the corresponding hole fieldsand their transformation properties. In the following procedure there are again parallels tothe case of the Dirac fermions, but it is a little more complicated because of the non-linearrealization of the SU(2)s symmetry. A priori, the fermion operators or fields we will use inthe following sections describe a combination of electrons and holes, which is why we referthem to as fermion operators or fields, respectively. The finally used hole fields are identifiedin the last section of this chapter.

6.1 Fermion Operators with a Sublattice Index

In [10] the single hole sector of the t-J model is investigated with a loop-cluster algorithm.There the energy-momentum relation is simulated (fig. 6.1). It shows that at low energiesthe holes live at the corners of the first Brillouin zone, i.e. in the same pockets as the Diracfermions in the weak coupling limit. Hence, the effective theory has to describe holes living

0.4

0.6

0.8

1

1.2

1.4

1.6

Figure 6.1: Energy-momentum relation E(k)/t for a single hole in an antiferromagnet on thehoneycomb lattice in the t-J model for J/t = 2. (Figure taken from [10].)

35

36 6. From Microscopic Operators to Effective Fields for Holes

in these pockets. To address the pockets we use the same sublattices as in section 4.2.1.Also here, the connection between the microscopic operators and the Grassmann fields cannot be derived in a rigorous way because of the different nature of the microscopic theoryand the effective theory. Now we also have to take the spontaneous symmetry breaking intoaccount. As an intermediate step, we introduce fermionic lattice operators by multiplying thediagonalizing field u(x) of eq. (5.14) and the microscopic operator Cx of eq. (3.17)

ΨXx = u(x)Cx = u(x)

(

cx↑ (−1)x c†x↓cx↓ − (−1)x c†x↑

)

, (6.1)

where x ∈ X stands for a discrete lattice site andX ∈ A1, A2, A3, B1, B2, B3 for a sublattice.For the even A and odd B sublattices we have

ΨXx = u(x)

(

cx↑ c†x↓cx↓ −c†x↑

)

=

(

ψXx,+ ψX†

x,−ψXx,− −ψX†

x,+

)

, x ∈ X ∧X ∈ A1, A2, A3,

ΨXx = u(x)

(

cx↑ −c†x↓cx↓ c†x↑

)

=

(

ψXx,+ −ψX†

x,−ψXx,− ψX

†

x,+

)

, x ∈ X ∧X ∈ B1, B2, B3. (6.2)

Besides addressing the sublattices, this definition is the connection between the microscopicmodel and the effective theory. It also guarantees that the global SU(2)s transformationsare realized non-linearly in the subgroup U(1)s on the fermionic degrees of freedom (seeeq. (6.3)). Since SU(2)s is spontaneously broken, the spin along the x3-axis is not a goodquantum number any more. Instead we measure the spin of each fermion relative to thelocal staggered magnetization. Doing so, we indicate spin no longer with an arrow, but witha plus or minus sign, indicating parallel or antiparallel alignment with respect to the localstaggered magnetization. The operators ψXx± obey standard anticommutation relations. Usingthe transformation rules of u(x) and Cx one finds the ones for ΨX

x

SU(2)s : ΨXx

′= h(x)ΨX

x ,

SU(2)Q :~QΨX

x = ΨXx ΩT ,

Di : DiΨXx = ΨDiX

x+ai ,

O : OΨXx = τ(Ox)ΨOX

Ox σ3,

O′ : O′

ΨXx = (iσ2)Ψ

OXOx σ3,

R : RΨXx = ΨRX

Rx . (6.3)

DiX, OX, and RX are the sublattices after the corresponding transformation. Since we arenow dealing with fermions, here the SU(2)Q symmetry acts non-trivially. As for Cx, we donot list the transformation rule under time-reversal T and T ′ as this is implemented by anantiunitary operator. However, time-reversal will be implemented afterwards on the effectivefields.

6.2 Fermion Fields with a Sublattice Index

Again we change from the Hamiltonian to the Lagrangian formalism used in a Euclidean pathintegral. With this we change from the microscopic operators to Grassmann fields living in

6.2 Fermion Fields with a Sublattice Index 37

the space-time continuum. We connect the two formulations by identifying matrix-valuedfermion fields with the fermionic lattice operators of eq. (6.2), similar as we did for the Diracfermions

ΨX(x) =

(

ψX+ (x) ψX†− (x)

ψX− (x) −ψX†+ (x)

)

, X ∈ A1, A2, A3,

ΨX(x) =

(

ψX+ (x) −ψX†− (x)

ψX− (x) ψX†+ (x)

)

, X ∈ B1, B2, B3. (6.4)

Doing so, the matrix-valued fermion fields inherit the transformation rules from the micro-scopic operators. The conjugate matrix-valued fields read

ΨX†(x) =

(

ψX†+ (x) ψX†

− (x)ψX− (x) −ψX+ (x)

)

, X ∈ A1, A2, A3,

ΨX†(x) =

(

ψX†+ (x) ψX†

− (x)−ψX− (x) ψX+ (x)

)

, X ∈ B1, B2, B3. (6.5)

To avoid confusion with relativistic theories, we do not denote the conjugate fields by ψX± (x).In contrast to their components (Grassmann fields), the matrix-valued fields ΨX†(x) andΨX(x) are not independent as they consist of the same fields. For the matrix-valued fermionfields we have the following transformation rules

SU(2)s : ΨX(x)′ = h(x)ΨX(x),

SU(2)Q :~QΨX(x) = ΨX(x)ΩT ,

Di : DiΨX(x) = ΨDiX(x),

O : OΨX(x) = τ(Ox)ΨOX(Ox)σ3,

O′ : O′

ΨX(x) = (iσ2)ΨOX(Ox)σ3,

R : RΨX(x) = ΨRX(Rx),

T : TΨX(x) = τ(Tx)(iσ2)[

ΨX†(Tx)T]

σ3,

TΨX†(x) = −σ3

[

ΨX(Tx)T]

(iσ2)†τ(Tx)†,

T ′ : T ′

ΨX(x) = −[

ΨX†(Tx)T]

σ3,

T ′

ΨX†(x) = σ3

[

ΨX(Tx)T]

. (6.6)

An upper index T on the right denotes transpose, while on the left it denotes time-reversal.Here we now list the transformation behaviour under time-reversal T and T ′. Time-reversalin the effective theory with nonlinearly realized SU(2)s symmetry follows from the usual formof time-reversal in the path integral of a nonrelativistic theory in which the spin symmetryis linearly realized. The fermion fields in the two formulations just differ by a factor u(x).

38 6. From Microscopic Operators to Effective Fields for Holes

From the transformation rules above one can read off the ones of the components

SU(2)s : ψX± (x)′ = exp(±iα(x))ψX± (x),

U(1)Q : QψX± (x) = exp(iω)ψX± (x),

Di : DiψX± (x) = ψDiX± (x),

O : OψX± (x) = ∓ exp(∓iϕ(Ox))ψOX∓ (Ox),

O′ : O′

ψX± (x) = ±ψOX∓ (Ox),

R : RψX± (x) = ψRX± (Rx),

T : TψX± (x) = exp(∓iϕ(Tx))ψX†± (Tx),

TψX†± (x) = − exp(±iϕ(Tx))ψX± (Tx),

T ′ : T ′

ψX± (x) = −ψX†± (Tx),

T ′

ψX†± (x) = ψX± (Tx). (6.7)

Since the SU(2)Q symmetry acts simply only on the matrix-valued fields, we only considerthe U(1)Q symmetry on the level of the component Grassmann fields. Under time-reversal,both the spin and the staggered magnetization change their sign. As a result, the projectionof the spin on the staggered magnetization does not change.

6.3 Fermion Fields with a Sublattice and Momentum Index

To describe the fermions living in the α and β pocket, we make the same discrete Fouriertransformation as for the Dirac fermions of eq. (4.19) with the matrix-valued fields of eq.(6.4). Then we obtain matrix-valued fermion fields with a momentum index composed of thesame linear combinations as we had for the Dirac fermions

ΨA,α(x) =1√3

(

exp(i2π3 )ΨA1(x) + ΨA2(x) + exp(−i2π3 )ΨA3(x))

,

ΨA,β(x) =1√3

(

exp(−i2π3 )ΨA1(x) + ΨA2(x) + exp(i2π3 )ΨA3(x))

,

ΨB,α(x) =1√3

(

exp(i2π3 )ΨB1(x) + ΨB2(x) + exp(−i2π3 )ΨB3(x))

,

ΨB,β(x) =1√3

(

exp(−i2π3 )ΨB1(x) + ΨB2(x) + exp(i2π3 )ΨB3(x))

. (6.8)

They take the form

ΨA,f(x) =

(

ψA,f+ (x) ψA,f′†

− (x)

ψA,f− (x) −ψA,f ′†+ (x)

)

, ΨB,f (x) =

(

ψB,f+ (x) −ψB,f ′†− (x)

ψB,f− (x) ψB,f′†

+ (x)

)

, (6.9)

and their conjugate counterparts are

ΨA,f†(x) =

(

ψA,f†+ (x) ψA,f†− (x)

ψA,f′

− (x) −ψA,f ′+ (x)

)

, ΨB,f†(x) =

(

ψB,f†+ (x) ψB,f†− (x)

−ψB,f ′− (x) ψB,f′

+ (x)

)

. (6.10)

6.3 Fermion Fields with a Sublattice and Momentum Index 39

Their transformation rules are

SU(2)s : ΨX,f (x)′ = h(x)ΨX,f (x),

SU(2)Q :~QΨX,f (x) = ΨX,f (x)ΩT ,

Di : DiΨX,f (x) = exp(ikfai)ΨX,f (x),

O : OΨA,α(x) = exp(−i2π3 )τ(Ox)ΨB,β(Ox)σ3,OΨA,β(x) = exp(i2π3 )τ(Ox)ΨB,α(Ox)σ3,

OΨB,α(x) = exp(i2π3 )τ(Ox)ΨA,β(Ox)σ3,OΨB,β(x) = exp(−i2π3 )τ(Ox)ΨA,α(Ox)σ3,

O′ : O′

ΨA,α(x) = exp(−i2π3 )(iσ2)ΨB,β(Ox)σ3,

O′

ΨA,β(x) = exp(i2π3 )(iσ2)ΨB,α(Ox)σ3,

O′

ΨB,α(x) = exp(i2π3 )(iσ2)ΨA,β(Ox)σ3,

O′

ΨB,β(x) = exp(−i2π3 )(iσ2)ΨA,α(Ox)σ3,

R : RΨX,f (x) = ΨX,f ′(Rx),

T : TΨX,f (x) = τ(Tx)(iσ2)[

ΨX,f ′†(Tx)T]

σ3,

TΨX,f†(x) = −σ3

[

ΨX,f ′(Tx)T]

(iσ2)†τ(Tx)†,

T ′ : T ′

ΨX,f (x) = −[

ΨX,f ′†(Tx)T]

σ3,

T ′

ΨX,f†(x) = σ3

[

ΨX,f ′(Tx)T]

. (6.11)

From them one can again read off the transformation rules for the components

SU(2)s : ψX,f± (x)′ = exp(±iα(x))ψX,f± (x),

U(1)Q : QψX,f± (x) = exp(iω)ψX,f± (x),

Di : DiψX,f± (x) = exp(ikfai)ψX,f± (x),

O : OψA,α± (x) = ∓ exp(−i2π3 ∓ iϕ(Ox))ψB,β∓ (Ox),

OψA,β± (x) = ∓ exp(i2π3 ∓ iϕ(Ox))ψB,α∓ (Ox),

OψB,α± (x) = ∓ exp(i2π3 ∓ iϕ(Ox))ψA,β∓ (Ox),

OψB,β± (x) = ∓ exp(−i2π3 ∓ iϕ(Ox))ψA,α∓ (Ox),

O′ : O′

ψA,α± (x) = ± exp(−i2π3 )ψB,β∓ (Ox),

O′

ψA,β± (x) = ± exp(i2π3 )ψB,α∓ (Ox),

O′

ψB,α± (x) = ± exp(i2π3 )ψA,β∓ (Ox),

O′

ψB,β± (x) = ± exp(−i2π3 )ψA,α∓ (Ox),

R : RψX,f± (x) = ψX,f′

± (Rx),

T : TψX,f± (x) = exp(∓iϕ(Tx))ψX,f′†

± (Tx),

TψX,f†± (x) = − exp(±iϕ(Tx))ψX,f′

± (Tx),

T ′ : T ′

ψX,f± (x) = −ψX,f ′†± (Tx),

T ′

ψX,f†± (x) = ψX,f′

± (Tx). (6.12)

40 6. From Microscopic Operators to Effective Fields for Holes

6.4 Hole Fields

As already mentioned in the introduction of this chapter, the fermion fields may describe acombination of electron and hole fields. In this section we will now identify the final holefields. Using the matrix-valued fermion fields we are able to write down a Lagrangian whichis SU(2)Q invariant. However, this would be necessary only for the Hubbard model at half-filling, where electrons and holes are present at the same time. Real materials and also the t-Jmodel do not contain both. This means that the effective Lagrangian which we will constructonly has to be U(1)Q invariant. Hence, we should break the SU(2)Q symmetry explicitly.The hole fields can be identified by first writing down all mass terms with the matrix-valuedfermion fields with momentum index of eq. (6.9). Second, after writing the mass terms inspinor notation, one has to diagonalize the obtained mass matrices. Then the entries in thediagonals are the masses of the fermions.

The trace notation used for the matrix-valued fermion fields has several difficulties. Onenever knows whether one has found all terms, it is much harder to see linear dependencies,and terms may be zero (for more details see [4]). Because of that, we also constructed allmass terms in component notation. From this we then knew that there are two linearlyindependent mass terms. They read

∑

f=α,β

1

2Tr[

M(ΨA,f†σ3ΨA,f − ΨB,f†σ3Ψ

B,f ) +m(ΨA,f†ΨA,fσ3 + ΨB,f†ΨB,fσ3)]

=∑

f=α,β

[

M(

ψA,f†+ ψA,f+ − ψA,f†− ψA,f− + ψB,f†− ψB,f− − ψB,f†+ ψB,f+

)

+m(

ψA,f†+ ψA,f+ + ψA,f†− ψA,f− + ψB,f†+ ψB,f+ + ψB,f†− ψB,f−)]

=∑

f=α,β

[

(

ψA,f†+ ψB,f†+

)

(

M +m 00 −M +m

)(

ψA,f+

ψB,f+

)

+(

ψA,f†− , ψB,f†−)

(

−M +m 00 M +m

)(

ψA,f−ψB,f−

)]

. (6.13)

The term proportional to M is invariant under SU(2)Q transformations while the one propor-tional to m is only invariant under U(1)Q and breaks SU(2)Q explicitly. Since the obtainedmatrices are already diagonal, we don’t have to diagonalize them. Without the SU(2)Q-breaking term (i.e. for m = 0), the eigenvalues of the mass matrices, and with this the massesof the fermions, are ±M and we have a perfect symmetry. When we also take the SU(2)Q-breaking term into account, these masses are shifted to ±M +m. The hole fields correspondto the lower eigenvalue −M +m and are identified as the corresponding eigenvectors

ψB,α+ (x), ψB,β+ (x), ψA,α− (x), ψA,β− (x). (6.14)

In contrast to the square lattice case, where the mass matrices are not automatically diagonal,we do not obtain linear combinations but only identify the four correct fields. As we removethe electrons, the electron-hole symmetry is broken explicitly and the Lagrangian describingdoped holes has to be invariant only under the remaining U(1)Q fermion number symmetry.However, the SU(2)Q symmetry was important to identify the hole fields. Electron fieldscould be identified in the same way as the hole fields. The notation of the four remaining

6.4 Hole Fields 41

fields can be simplified as B is always combined with + and A with −. Thus in the followingwe drop the sublattice index and we obtain the final hole fields

ψα+(x) = ψB,α+ (x), ψβ+(x) = ψB,β+ (x), ψα−(x) = ψA,α− (x), ψβ−(x) = ψA,β− (x), (6.15)

and their conjugated counterparts. Finally, we list their transformation rules, which are thesame as in eq. (6.12), but with the simpler notation

SU(2)s : ψf±(x)′ = exp(±iα(x))ψf±(x),

U(1)Q : Qψf±(x) = exp(iω)ψf±(x),

Di : Diψf±(x) = exp(ikfai)ψf±(x),

O : Oψα±(x) = ∓ exp(±i2π3 ∓ iϕ(Ox))ψβ∓(Ox),

Oψβ±(x) = ∓ exp(∓i2π3 ∓ iϕ(Ox))ψα∓(Ox),

O′ : O′

ψα±(x) = ± exp(±i2π3 )ψβ∓(Ox),

O′

ψβ±(x) = ± exp(∓i2π3 )ψα∓(Ox),

R : Rψf±(x) = ψf′

± (Rx),

T : Tψf±(x) = exp(∓iϕ(Tx))ψf′†

± (Tx),

Tψf†± (x) = − exp(±iϕ(Tx))ψf′

± (Tx),

T ′ : T ′

ψf±(x) = −ψf ′†± (Tx),

T ′

ψf†± (x) = ψf′

± (Tx). (6.16)

They are mapped on themselves under the various symmetry transformations and the spon-taneously broken SU(2)s symmetry is realized in a non-linear way on the hole fields.

Chapter 7

Effective Field Theory for Magnons

and Holes

In chapter 5 we derived the effective magnon action for an undoped antiferromagnet. A pureantiferromagnet is the ground state or vacuum of our theory and corresponds to the half-filledsystem in the Hubbard model. It is like the Dirac sea in a relativistic quantum field theoryand it is a charge neutral state. When we remove electrons from the half-filled system, weget doped holes. In this chapter we want to extend the action to describe also doped holesin addition to the magnons. We do this again in a similar way as for the Dirac fermions. Atthe end we discuss some accidental emergent symmetries.

7.1 Effective Action for Magnons and Holes

One can write the effective action for magnons and holes as

S[

ψf†± , ψf±, ~e]

=

∫

d2x dt∑

nψ

Lnψ , (7.1)

where nψ stands for the number of hole fields. The partition function takes the form

Z =

∫

Dψf†± Dψf±D~e exp(

−S[

ψf†± , ψf±, ~e])

. (7.2)

For completeness we again write down the leading terms of the low-energy effective magnonLagrangian

L0 =ρs2

(

∂i~e · ∂i~e+1

c2∂t~e · ∂t~e

)

= 2ρs

(

v+i v

−i +

1

c2v+t v

−t

)

. (7.3)

Before we can begin with the construction of the effective Lagrangian describing holes, whichconsists of derivatives, hole fields and the magnon fields v3

µ(x) and v±µ (x), we have to considerhow the number of derivatives is counted. As mentioned, the magnons have a relativisticenergy-momentum relation (E ∝ p) at low energies. Therefore, both, spatial and temporalderivatives on magnon fields correspond to first order. The holes on the other side, have a non-relativistic energy-momentum relation (E ∝ p2) at low energies. Thus a temporal derivativeacting on a hole filed corresponds to second order while a spatial derivative corresponds tofirst order. This causes problems as derivatives can be shifted around by partial integration.For example a temporal derivative may act first on a hole field and after a partial integration

43

44 7. Effective Field Theory for Magnons and Holes

on a magnon field. To solve this problem completely, which is necessary when one wants todo loop-calculations, one would have to develop a power counting scheme. This is nontrivialand since we are only interested in the leading order, we don’t need it for our purposes.Therefore we won’t develop such a power counting scheme at the moment. Instead we makethe convention that we always let temporal derivatives act on the hole fields if such exist ina given term. In the following we will go up to second order in the expansion to also includetemporal derivatives.

One has to keep in mind, that the magnon fields also contain a derivative. As we did inthe case of the Dirac fermions, we characterize the parts of the Lagrangian by the number ofhole fields they contain. We argued that always as many daggered as undaggered hole fieldsare needed in a term. Up to second order, the contribution L2, which describes a single α orβ-hole doped in the antiferromagnet and its interactions with magnons reads

L2 =∑

f=α,βs=+,−

[

Mψf†s ψfs + ψf†s Dtψ

fs +

1

2M ′Diψf†s Diψ

fs

+ Λψf†s (isvs1 + σfvs2)ψ

f−s + iK

[

(D1 + isσfD2)ψf†s (vs1 + isσfv

s2)ψ

f−s

− (vs1 + isσfvs2)ψ

f†s (D1 + isσfD2)ψ

f−s]

+ σfLψf†s ǫijf

3ijψ

fs

+N1ψf†s v

si v

−si ψfs + isσfN2

(

ψf†s vs1v

−s2 ψfs − ψf†s v

s2v

−s1 ψfs

)

]

, (7.4)

with the field strength tensor of the composite Abelian ”gauge” field defined by

f3ij(x) = ∂iv

3j (x) − ∂jv

3i (x). (7.5)

σf is still + for f = α and − for f = β. As already mentioned, the SU(2)s symmetry is notreally gauged, it just acts on the hole fields as a local U(1)s gauge-like symmetry. To makethe Lagrangian invariant, we introduced the covariant derivatives

Dtψf±(x) =

[

∂t ± iv3t (x) − µ

]

ψf±(x),

Diψf±(x) =

[

∂i ± iv3i (x)

]

ψf±(x),

Dtψf†± (x) =

[

∂t ∓ iv3t (x) − µ

]

ψf†± (x),

Diψf†± (x) =

[

∂i ∓ iv3i (x)

]

ψf†± (x). (7.6)

The chemical potential µ enters the covariant time-derivative like an imaginary constantvector potential for the fermion number symmetry U(1)Q.

All low-energy constants in L2 are real. M is the rest mass and M ′ the kinetic mass ofa hole. Since we deal with a non-relativistic theory, they don’t have to be the same. Termsincluding magnon fields describe hole-magnon interactions. As we will see in chapter 8, v±µ (x)corresponds to one magnon and v3

µ(x) to two magnons. The terms proportional to Λ are theleading hole-one-magnon interaction terms and are thus dominant. There is an M ′′-term butno L-term in the square lattice case. In fact, the two terms are similar but because of a minussign in case of the L-term, some contributions cancel and it can be written in a simpler form.The factor i in front of the K and N2-term ensures that the corresponding Hamiltonian isHermitian. Also here ρs defines the energy scale below which the low-energy expansion isvalid, i.e. for energy values small compared to ρs. In addition to the spin stiffness ρs and thespin wave velocity c (see tab. 5.1), also the value of the kinetic mass M ′ has been determinedin [10]. Its value is 4.1(1)/ta2, with t from the t-J model.

From eq. (7.4) one can derive a single hole energy-momentum relation

Ef (p) = M +p2i

2M ′ + O(p4), (7.7)

7.2 Accidental Emergent Symmetries 45

where the momentum is relative to the center of the corresponding hole pocket. This is theenergy-momentum relation of a free, nonrelativistic particle and it describes circular shapedhole pockets, which is in agreement with the simulation mentioned in section 6.1.

Terms with more than two hole fields are contact terms and therefore describe short-rangeinteractions. Because we are most interested in the long-range interactions, we only constructL4 without derivatives in addition to L2. It reads

L4 =∑

s=+,−

[G1

2(ψα†s ψ

αs ψ

α†−sψ

α−s + ψβ†s ψ

βs ψ

β†−sψ

β−s)

+G2ψα†s ψ

αs ψ

β†s ψ

βs +G3ψ

α†s ψ

αs ψ

β†−sψ

β−s

]

. (7.8)

All low-energy constants are real. As already mentioned in the case of the Dirac fermions,terms with identical fermion fields vanish due to the Pauli principle and thus also terms withmore than eight fermion fields and without derivatives again do not exist.

7.2 Accidental Emergent Symmetries

The fundamental forces underlying condensed matter physics are Poincare-invariant. By theformation of a crystal lattice, some of the space-time symmetries are spontaneously broken.The Goldstone bosons resulting from this spontaneous symmetry breaking are the phonons.They arise in the low-energy spectrum, but it is expected that they alone can not providea mechanism for Cooper pair formation. In the Hubbard model, and hence in our effectivetheory, these symmetries are broken explicitly as the lattice is given from the outset. Thereforethe Hubbard model and the effective theory don’t contain phonons. In this context of a non-relativistic theory, in addition to the breakdown of the continuous translation and rotationsymmetries to their discrete counterparts, also the Galilean boost invariance is broken. Thisprovides a preferred rest frame and because of that, magnon-mediated forces between a pairof holes may depend on the center of mass momentum of the pair.

If phonons were important, one could construct an effective theory of spontaneously brokenSU(2)s and Galilean symmetry, which then would include magnons and phonons. We assumein this thesis that phonons do not play an important role. Hence it is legitimate to break theGalilean symmetry by hand, thus excluding phonons.

In the following two sections we show that some leading terms of the effective Lagrangianshow accidentally emergent symmetries which are broken by higher orders in the expansion.

7.2.1 Galilean Boost Symmetry G

Ignoring the higher order K-term, the L2 (of eq. (7.4)) shows an accidental Galilean boostsymmetry G acting on the different fields as

G : GP (x) = P (Gx), Gx = (x1 − v1t, x2 − v2t, t),

Gψf±(x) = exp(−pfi xi + ωf t)ψf±(Gx),

Gψf†± (x) = exp(pfi xi − ωf t)ψf†± (Gx),Gv3

i (x) = v3i (Gx),

Gv3t (x) = v3

t (Gx) − viv3i (Gx),

Gv±i (x) = v±i (Gx),Gv±t (x) = v±t (Gx) − viv

±i (Gx), (7.9)

46 7. Effective Field Theory for Magnons and Holes

with

pf1 = M ′v1, pf2 = M ′v2, (7.10)

and the energy-momentum relation

ωf =(pfi )

2

2M ′ . (7.11)

Here the vi denote the boost velocity. By means of this accidental Galilean boost symmetry,the magnon-mediated forces between two holes don’t depend on the total momentum and itis possible to investigate the hole pair in their rest frame without loss of generality as long asonly the leading Λ-term is involved (see chapter 9).

7.2.2 Continuous O(γ) Rotation Symmetry

Again ignoring the higher order K-term, L2 also has an accidental continuous O(γ) rotationsymmetry which acts on the fields as

O(γ)ψf±(x) = exp(isσfγ2 )ψf±(O(γ)x),

O(γ)v1(x) = cos γ v1(O(γ)x) + sin γ v2(O(γ)x),

O(γ)v2(x) = − sin γ v1(O(γ)x) + cos γ v2(O(γ)x), (7.12)

withO(γ)x = O(γ)(x1, x2, t) = (cos γ x1 − sin γ x2, sin γ x1 + cos γ x2, t). (7.13)

Here the vi stand for the composite magnon field of eq. (5.23). The Λ-term also has thissymmetry. This implies that, in contrast to the square lattice case, where no such accidentalrotation symmetry occurs, spirals in the staggered magnetization can point in an arbitrarydirection. This is discussed in [9, 11].

Chapter 8

One-Magnon Exchange Potentials

At low energies, the long-range dynamics is dominated by one-magnon exchange betweenholes. Since we are interested in possible long-range attractive forces between hole pairsas potential mechanisms for Cooper pair formation, we will focus on them. Hence, in thefollowing we are going to use the effective theory to investigate the one-magnon exchangebetween a pair of holes. For this purpose we first derive the one-magnon exchange potentialsbetween an isolated pair of holes in an otherwise undoped antiferromagnet in this chapter.The system is considered undoped as two additional holes do not cause a finite hole density.It is also possible to investigate a system with finite hole density, which is done in [9, 11] inthe context of spiral phases. In the following by ”one-magnon physics” we mean processes inwhich a single magnon couples to holes. After deriving the potentials, in the next chapter wewill write down the corresponding Schrodinger equation and discuss it. In this chapter we dealwith two-dimensional space vectors and three-dimensional space-time vectors. To distinguishthem, from now on two-dimensional space vectors are denoted with a vector arrow while thethree-dimensional space-time vectors are not.

8.1 One-Magnon Action

Aiming at the potentials, we will first derive the one-magnon action, which is the reducedaction only describing a single magnon propagating and coupling to holes. For this we nowdo some preparation. In the ground state of an antiferromagnet, the staggered magnetizationpoints in the same direction at every space-point. Without loss of generality, we can choose theordered configuration to point in the x3-direction, i.e. ~e(x) = (0, 0, 1). To include magnons,we allow small fluctuations in the local staggered magnetization. We expand the magnonfluctuations m1(x) and m2(x) around the ordered staggered magnetization up to first order

~e(x) =

(

m1(x)√ρs

,m2(x)√ρs

, 1

)

+ O(

m2)

. (8.1)

The L0 from eq. (7.3) expanded in the magnon fluctuations reads

L0 =1

2

(

∂ima∂ima +1

c2∂tma∂tma

)

+ O(

m4)

, (8.2)

with a sum over the repeated indices. It describes the propagation of a magnon, thereforecontains two fields, and is thus quadratic in the fluctuations. The magnons are coupled to the

47

48 8. One-Magnon Exchange Potentials

holes through the v3µ(x) and v±µ (x) fields. We also expand them in the magnon fluctuations

v±µ (x) =1

2√ρs∂µ[

m2(x) ± im1(x)]

+ O(

m3)

,

v3µ(x) =

1

4ρs

[

m1(x)∂µm2(x) −m2(x)∂µm1(x)]

+ O(

m4)

. (8.3)

One can see that v±µ (x) is linear in the fluctuations and thus describes one magnon, whilev3µ(x) is quadratic and describes two magnons. Using the combination

m(x) = m1(x) + im2(x), (8.4)

we can write them in a more compact form

v+µ (x) =

i

2√ρs∂µm

∗(x), v−µ (x) = − i

2√ρs∂µm(x). (8.5)

Now we will extract the one-magnon physics from L0 and L2 (eqs. (7.3) and (7.4)). Thus,apart from the propagator, we only take contributions up to first order in the fluctuations intoaccount. This means that the v3

µ(x) fields drop out. The low-energy effective action relevantfor one-magnon exchange then reads

S[ψf†± ,ψf±,m] =

∫

d2x dt

1

2

(

∂im∗∂im+

1

c2∂tm

∗∂tm

)

− 1

2√ρs

∑

f=α,β

[

Λ(

ψf†+ (∂1m∗ + iσf∂2m

∗)ψf− + ψf†− (∂1m− iσf∂2m)ψf+

)

+K(

(∂1 − iσf∂2)ψf†+ (∂1m

∗ − iσf∂2m∗)ψf−

− (∂1m∗ − iσf∂2m

∗)ψf†+ (∂1 − iσf∂2)ψf− − (∂1 + iσf∂2)ψ

f†− (∂1m+ iσf∂2m)ψf+

+ (∂1m+ iσf∂2m)ψf†− (∂1 + iσf∂2)ψf+

)

]

+ O(m2)L2+ O(m4)L0

. (8.6)

We dropped the terms describing only holes. From the hole-magnon interaction terms only theΛ and parts of the K-term describe one-magnon couplings. One can see that a hole undergoesa spin flip by emitting or absorbing a magnon. The magnon carries the spin difference one inthe sense of a flavour quantum number. This implies that only holes with opposite spin canexchange a magnon.

8.2 Kinematics

We now discuss the kinematics of one-magnon exchange at tree level. The correspondingFeynman diagram is shown in fig. 8.1. The ~p± are the momenta of the incoming and the ~p±′

the momenta of the outgoing holes, while the momentum carried by the exchanged magnonis denoted by ~q. Here f = α, β and f = α, β stand for the flavour of the incoming spin +and − hole, respectively. The incoming and outgoing holes are asymptotically free with theenergy-momentum relation from eq. (7.7)

Ef (~p ) = M +p2i

2M ′ + O(~p 4). (8.7)

8.3 Transforming to Momentum Space 49

f+

f−

f−

f+

~p+

~q

~p′

−

~p− ~p′

+

Figure 8.1: Tree-level Feynman diagram for one-magnon exchange between two holes. (Figuretaken from [5].)

Momentum conservation at the vertices implies

~p+ = ~q + ~p−′ , and ~p− = ~p+

′ − ~q, ⇒ ~p+ + ~p− = ~p+′ + ~p−

′ , (8.8)

where we can identify the total momentum ~P of the hole pair

~P = ~p+ + ~p− = ~p+′ + ~p−

′ . (8.9)

Energy conservation at the vertices implies

Ef (~p+) = q0 + Ef (~p−′), and Ef (~p−) = Ef (~p+

′) − q0, (8.10)

where q0 stands for the energy of the magnon. Solving these two equations for q0 gives

q0 =1

2M ′ (~p+2 − ~p−

′2), q0 =1

2M ′ (~p+′2 − ~p−

2). (8.11)

In the following we will use the energy-momentum vectors

p =(

~p ,Ef (~p ))

,

q = (~q , q0). (8.12)

The actual form of q0 will not be used in the following. From the eqs. (8.11) we only want tonote that q0 ∝ ~p 2. The following calculations will always involve q2, never just q. Because q20is proportional to ~p 4 and since we work only up to order ~p 2, q2 and ~q 2 are actually the same.

8.3 Transforming to Momentum Space

It is convenient to work in momentum space. Therefore we will transform the action relevantfor one-magnon physics from eq. (8.6) into momentum space. The Fourier transformations ofthe hole fields and the magnon fluctuations take the form

ψf±(x) =1

(2π)3

∫

d3pψf±(p) exp(−ipx),

ψf†± (x) =1

(2π)3

∫

d3pψf†± (p) exp(ipx),

m(x) =1

(2π)3

∫

d3q m(q) exp(iqx),

m∗(x) =1

(2π)3

∫

d3q m∗(q) exp(−iqx). (8.13)

50 8. One-Magnon Exchange Potentials

With these we can construct the combinations arising in the action

∫

d3x ∂µm∗(x)∂µm(x) =

1

(2π)3

∫

d3q m(q)m∗(q)q2µ, (no sum over µ),

∫

d3xψf†+ (x)∂jm∗(x)ψf−(x) =

−i(2π)6

∫

d3p− d3q ψf†+ (p− + q)m∗(q)ψf−(p−)qj,

∫

d3xψf†− (x)∂jm(x)ψf+(x) =i

(2π)6

∫

d3p+ d3q ψf†− (p+ − q)m(q)ψf+(p+)qj,

∫

d3x ∂iψf†+ (x)∂jm

∗(x)ψf−(x) =1

(2π)6

∫

d3p− d3q ψf†+ (p− + q)m∗(q)ψf−(p−)qj(p−i + qi),

∫

d3x ∂im∗(x)ψf†+ (x)∂jψ

f−(x) = − 1

(2π)6

∫

d3p− d3q ψf†+ (p− + q)m∗(q)ψf−(p−)p−jqi,

∫

d3x ∂iψf†− (x)∂jm(x)ψf+(x) = − 1

(2π)6

∫

d3p+ d3q ψf†− (p+ − q)m(q)ψf+(p+)qj(p+i − qi),

∫

d3x ∂im(x)ψf†− (x)∂jψf+(x) =

1

(2π)6

∫

d3p+ d3q ψf†− (p+ − q)m(q)ψf+(p+)p+jqi. (8.14)

By means of these relations we obtain the one-magnon action in momentum space

S[ψf†± ,ψf±,m] =

∫

d3q∑

f=α,β

1

(2π)31

2m∗mq2

+1

(2π)6Λ

2√ρs

[ ∫

d3p− ψf†+ m∗ ψf−

(

iq1 + σfq2)

−∫

d3p+ ψf†− mψf+

(

iq1 − σfq2)

]

+1

(2π)6iK

2√ρs

[ ∫

d3p− ψf†+ m∗ ψf−

(

iq1(p−1 + q1) − iq2(p−2 + q2)

− σf(

q1(p−2 + q2) + q2(p−1 + q1) + p−1q2 + p−2q1)

+ ip−1q1 − ip−2q2

)

+

∫

d3p+ ψf†− mψf+

(

iq1(p+1 − q1) − iq2(p+2 − q2)

+ σf(

q1(p+2 − q2) + q2(p+1 − q1) + p+1q2 + p+2q1)

+ ip+1q1 − ip+2q2

)

]

. (8.15)

The term proportional to 1/c2 in the magnon propagator disappeared because it is propor-tional to p2

0 and thus proportional to ~p 4. Note that the arguments of the fields are alwaysthe same and that we drop them when writing down a Lagrangian.

8.4 Principle of Stationary Action

Since we consider the magnon exchange only at tree level, the physics is of non-quantumnature and thus we can use the principle of stationary action. Therefore we extremize theaction in momentum space with respect to the magnon fluctuations m∗(q) and m(q) with the

8.5 Potentials in Momentum Space 51

help of the Euler-Lagrange equations. This results in

m(q) = − 1

(2π)3√ρsq2

∫

d3p−∑

f=α,β

Λψf†+ (p− + q)ψf−(p−)(

iq1 + σfq2)

−Kψf†+ (p− + q)ψf−(p−)(

q1 + iσfq2)

×(

q1 + 2p−1 + iσf (2p−2 + q2))

, (8.16)

and

m∗(q) =1

(2π)3√ρsq2

∫

d3p+

∑

f=α,β

Λψf†− (p+ − q)ψf+(p+)(

iq1 − σfq2)

−Kψf†− (p+ − q)ψf+(p+)(

q1 − iσfq2)

×(

q1 − 2p+1 + iσf (2p+2 − q2))

. (8.17)

When we plug the obtained expressions for m∗(q) and m(q) back into the action, the magnonfields are integrated out and we obtain

S[ψf†± ,ψf±]stat = − 1

(2π)9

∫

d3q d3p+ d3p−

∑

f=α,β

1

2ρs

Λ2

[

ψf†+ ψf−ψf†− ψ

f+ − ψf†+ ψf−ψ

f ′†− ψf

′

+

1

q2(

iq1 + σfq2)2]

+K2

[

ψf†+ ψf−ψf†− ψ

f+

(

2p+1 − q1 − iσf (2p+2 − q2))(

2p−1 + q1 + iσf (2p−2 + q2))

+1

q2ψf†+ ψf−ψ

f ′†− ψf

′

+

(

q1 + iσfq2)2(

2p+1 − q1 + iσf (2p+2 − q2))(

2p−1 + q1 + iσf (2p−2 + q2))

]

−KΛ

[

1

q2ψf†+ ψf−ψ

f†− ψ

f+

(

q1 − iσfq2)2(

i(2p+1 − q1) + σf (2p+2 − q2))

+ψf†+ ψf−ψf ′†− ψf

′

+

(

i(2p+1 − q1) − σf (2p+ − q2))

]

+ΛK

[

1

q2ψf†+ ψf−ψ

f†− ψ

f+

(

q1 + iσfq2)2(

i(2p−1 + q1) − σf (2p−2 + q2))

+ψf†+ ψf−ψf ′†− ψf

′

+

(

i(2p−1 + q1) − σf (2p−2 + q2))

]

. (8.18)

The ordering of the low-energy constants indicates which term (Λ or K) is responsible for theupper and the lower vertex, respectively, in the Feynman diagram in fig. 8.1. Note that thesign of the exchange potentials (see next section) depends on the ordering of the hole fields.

The above ordering, i.e. ψf†+ ψf−ψf†− ψ

f+, leads to the correct over-all sign.

8.5 Potentials in Momentum Space

From the stationary action we can read off the resulting potentials for the various combina-tions. In momentum space they take the form

〈~p+′ ~p−

′|V ffFG|~p+~p−〉 = V ff

FG(~q ) δ(~p+ + ~p− − ~p+′ − ~p−

′), F,G ∈ Λ,K , (8.19)

52 8. One-Magnon Exchange Potentials

with

V ffΛΛ(q) = − Λ2

2ρs, V ff ′

ΛΛ (q) =Λ2

2ρsq2(

iq1 − σfq2)2,

V ffKK(q) = −K

2

2ρs

(

2(p+1 − iσfp+2) − q1 + iσfq2)(

2(p−1 + iσfp−2) + q1 + iσfq2)

,

V ff ′

KK(q) = − K2

2ρsq2(

q1 − iσfq2)2(

2(p+1 − iσfp+2) − q1 + iσfq2)(

2(p−1 − iσfp−2) + q1 − iσfq2)

,

V ffΛK(q) = − iΛK

2ρsq2(

q1 + iσfq2)2(

2(p−1 + iσfp−2) + q1 + iσfq2)

,

V ff ′

ΛK (q) = − iΛK2ρs

(

2(p−1 − iσfp−2) + q1 − iσfq2)

,

V ffKΛ(q) =

iKΛ

2ρsq2(

q1 − iσfq2)2(

2(p+1 − iσfp+2) − q1 + iσfq2)

,

V ff ′

KΛ (q) =iKΛ

2ρs

(

2(p+1 − iσfp+2) − q1 + iσfq2)

. (8.20)

Here the first low-energy constant and the first flavour correspond to the upper vertex, whilethe second low-energy constant and the second flavour correspond to the lower vertex in theFeynman diagram in fig. 8.1.

8.6 Potentials in Position Space

We already noted that the Λ-term is the leading contribution in one-magnon interactions. Inthe next chapter we will investigate the Schrodinger equation for the relative motion of twoholes. For this we will only use the leading contribution, i.e. only the ΛΛ-potentials. Hence,we have to transform these back into position space. The other potentials could also betransformed into position space. This is a little more complicated but can be done along thelines of [1]. For the transformation of the potentials with mixed flavours we use the integrals

∫

d2~qq2j~q 2

exp(i~q · ~x) = 2π(1

~x 2− 2

x2j

~x 4) + fδ(2)(~x), (no sum over j),

∫

d2~qq1q2~q 2

exp(i~q · ~x) = −4πx1x2

~x 4+ gδ(2)(~x), f, g ∈ R, (8.21)

which can be solved by using1

~q 2=

∫ ∞

0dλ exp(−λ~q 2). (8.22)

Then the ΛΛ-potentials in position space read

〈~r+′~r−′|V ffΛΛ |~r+~r−〉 = V ff

ΛΛ(~r ) δ(2)(~r+ − ~r−′) δ(2)(~r− − ~r+

′), (8.23)

with

V ffΛΛ(~r ) = − Λ2

2ρsδ(2)(~r ), V ff ′

ΛΛ (~r ) =Λ2

2πρs~r 2exp(2iσfϕ). (8.24)

The V ff ′

ΛΛ (~r ) potentials would also have additional short-distance δ-function contributionsresulting from eq. (8.21), which we dropped. Since we will model the short-distance repulsion

8.6 Potentials in Position Space 53

by a hard core radius, δ-function contributions will not be used in the following. Therefore wewill also not use the V ff

ΛΛ(~r ) potentials. All these contributions add to the 4-fermion contactinteractions. Here ~r = ~r+ − ~r− denotes the distance vector between the two holes and ϕis the angle between ~r and the x1-axis. The δ-functions in eq. (8.23) assure that the holesdo not change their position during the magnon exchange. Note that although the magnonstravel with the finite speed c, the potentials are instantaneous. Retardation effects occur onlyat higher orders. As already mentioned in section 7.2, because of the accidental emergentGalilean boost symmetry, the ΛΛ-potentials do not depend on the total momentum ~P of the

hole pair. The V ff ′

ΛΛ (~r ) potentials fall off with 1/r2.

Chapter 9

Two Hole Bound States

In this chapter we investigate the Schrodinger equation for the relative motion of two holes.

As mentioned in the last chapter, we only take the leading long-ranged V ff ′

ΛΛ (~r ) potentialsinto account. The short-distance forces we take into account by imposing a hard core radius,i.e. a boundary condition on the wave function near the origin. We derive the radial solutionfor a special case analytically and find that the magnon-mediated forces lead to bound statesof holes under a condition on some low-energy constants.

9.1 Corresponding Schrodinger Equation

Since the Λ-term shows an accidental Galilean boost symmetry, we can investigate the holepair in its rest frame without loss of generality. The kinetic energy of the hole pair is

T =∑

f=α,β

p2i

2M ′ + O(~p 4) =p2i

M ′ + O(~p 4). (9.1)

We introduce the two probability amplitudes Ψ1(~r ) and Ψ2(~r ) which represent the twoflavour-spin combinations α+β− and α−β+, respectively, where we choose the distance vec-tor ~r to point from the β to the α hole. Because the holes undergo a spin flip during themagnon exchange, the two probability amplitudes are coupled through the magnon exchangepotentials and the Schrodinger equation describing the relative motion of the hole pair is atwo-component equation which reads

(

− 1M ′∆ V βα

ΛΛ (~r )

V αβΛΛ (~r ) − 1

M ′∆

)

(

Ψ1(~r )Ψ2(~r )

)

= E

(

Ψ1(~r )Ψ2(~r )

)

, (9.2)

with the binding energy E. When we plug in the potentials, we obtain

( − 1M ′∆ γ 1

~r 2 exp(−2iϕ)

γ 1~r 2 exp(2iϕ) − 1

M ′∆

)(

Ψ1(~r )Ψ2(~r )

)

= E

(

Ψ1(~r )Ψ2(~r )

)

, (9.3)

with

γ =Λ2

2πρs. (9.4)

55

56 9. Two Hole Bound States

Since we are mostly interested in the radial part of the solution, we make the separationansatz

Ψ1(r, ϕ) = R1(r) exp(im1ϕ), Ψ2(r, ϕ) = R2(r) exp(im2ϕ), (9.5)

with r = |~r |. Inserting the ansatz, we get the two coupled equations

− 1

M ′∆(

R1(r) exp(im1ϕ))

+ γR2(r)

r2exp(−2iϕ) exp(im2ϕ) = ER1(r) exp(im1ϕ),

− 1

M ′∆(

R2(r) exp(im2ϕ))

+ γR1(r)

r2exp(2iϕ) exp(im1ϕ) = ER2(r) exp(im2ϕ). (9.6)

By using the Laplace operator in polar coordinates

∆Ψ(~r) =∂2Ψ(~r)

∂r2+

1

r

∂Ψ(~r)

∂r+

1

r2∂2Ψ(~r)

∂ϕ2, (9.7)

and after a multiplication of the first equation with M ′ exp(−im1ϕ) and of the second withM ′ exp(−im2ϕ), we obtain

−(

d2

dr2+

1

r

d

dr− 1

r2m2

1

)

R1(r) + γM ′R2(r)

r2exp

(

− iϕ(2 +m1 −m2))

= M ′ER1(r),

−(

d2

dr2+

1

r

d

dr− 1

r2m2

2

)

R2(r) + γM ′R1(r)

r2exp

(

iϕ(2 +m1 −m2))

= M ′ER2(r). (9.8)

To separate the radial from the angular equations the relation

2 +m1 −m2 = 0 (9.9)

has to be fulfilled. Therefore we introduce the parameter m

m1 = m− 1, m2 = m+ 1. (9.10)

Then we obtain the radial equations

−(

d2

dr2+

1

r

d

dr− 1

r2(m− 1)2

)

R1(r) + γM ′R2(r)

r2= M ′ER1(r),

−(

d2

dr2+

1

r

d

dr− 1

r2(m+ 1)2

)

R2(r) + γM ′R1(r)

r2= M ′ER2(r). (9.11)

In this thesis we content ourselves with the case m = 0, for which the two equations decoupleand can be solved analytically. The other cases would have to be investigated numerically.Now with m = 0, we first add the second equation to the first and then subtract the secondfrom the first. Then we obtain the two equations

[

−(

d2

dr2+

1

r

d

dr

)

+ (1 + γM ′)1

r2

]

(

R1(r) +R2(r))

= M ′E(

R1(r) +R2(r))

,

[

−(

d2

dr2+

1

r

d

dr

)

+ (1 − γM ′)1

r2

]

(

R1(r) −R2(r))

= M ′E(

R1(r) −R2(r))

. (9.12)

Because the two equations are different, but the energy E is the same, one of the equationshas a zero solution. In the first equation the potential always has a positive sign and is thus

9.1 Corresponding Schrodinger Equation 57

repulsive. In the second equation on the other hand, the potential has a negative sign and istherefore attractive when the low-energy constants obey the relation

1 − M ′Λ2

2πρs≤ 0. (9.13)

Thus, magnon-mediated forces can only lead to bound states if the low-energy constant Λ islarger than the critical value

Λc =

√

2πρsM ′ . (9.14)

The same critical value arises in the investigation of the spiral phases, see [9]. Hence we areinterested in the solution of the system where the first equation has a zero solution and thesecond a non-zero one. Then we have

R1(r) +R2(r) = 0. (9.15)

Therefore the second equation simplifies to[

−(

d2

dr2+

1

r

d

dr

)

+ (1 − γM ′)1

r2

]

R(r) = M ′ER(r), (9.16)

and we are now looking for the radial wave function R(r) between the two holes. The sameequation occurred in the square lattice case and we can solve it in the same way (see [1]). Itis ill-defined because the 1/r2 potential is too singular at the origin. However, we have notyet included the short-range interactions. We model the short-range repulsion by a hard coreradius r0, i.e. we require R(r0) = 0 for r ≤ r0.

Next we multiply the equation with −r2 and write E = − |E|, since we are now lookingfor bound states with E < 0. Thus we get

r2d2R(r)

dr2+ r

dR(r)

dr+(

γM ′ − 1 − r2M ′|E|)

R(r) = 0. (9.17)

By introducing the dimensionless parameters z =√

M ′ |E|r and ν = i√γM ′ − 1, we can bring

the equation in the form of a Bessel differential equation (see [44])

z2 d2R(z)

dz2+ z

dR(z)

dz−(

z2 + ν2)

R(z) = 0. (9.18)

This equation is solved by the modified Bessel function Kν(z). Thus we obtain the radialwave function of the two holes

R(r) = AKν

(√

M ′|E|r)

, ν = i√

γM ′ − 1, (9.19)

where A is a normalization constant. Demanding that the solution vanishes at the hard coreradius gives a quantization condition for the bound state energy. The quantum number nthen labels the n-th excited state. For large n, the binding energy is given by

En ∼ − 1

M ′r20exp

( −2πn√γM ′ − 1

)

. (9.20)

Like every quantity calculated with the effective theory, the binding energy depends on low-energy constants. The binding is exponentially small in n and there are infinitely many bound

58 9. Two Hole Bound States

states. While the highly excited states have exponentially small energy and exponentiallylarge size, for sufficiently small r0 the ground state could have a small size and be stronglybound. However, as already mentioned, for short-distance physics the effective theory shouldnot be trusted quantitatively. If the holes were really tightly bound, one could construct aneffective theory which incorporates them as explicit low-energy degrees of freedom. As longas the binding energy is small compared to the relevant high-energy scales such as ρs, ourresult is valid and receives only small corrections from higher-order effects.

Chapter 10

Conclusions and Outlook

In this thesis we developed a systematic low-energy effective field theory for magnons andholes in an antiferromagnet on the honeycomb lattice analogous to BχPT. As an underlyingmicroscopic model we used the Hubbard model. Apart from different values of the low-energyconstants, the effective theory remains the same for other underlying models which have thesame symmetries as the Hubbard model. We have shown that the leading action of the puremagnon sector is the same as in the square lattice case. The main part of this thesis wasthe construction of the leading terms of the effective Lagrangian describing magnons andholes. Most of the terms are similar to the ones in the square lattice case. However, there isan M ′′-term but no L-term in the square lattice case. In fact, the two terms are similar butbecause of a minus sign in case of the L-term, some contributions cancel and it can be writtenin a simpler form. There is also a second order K-term in case of the honeycomb lattice. Theimportant leading Λ-term also occurs. The leading terms of the effective Lagrangian show anaccidental emergent Galilean boost symmetry and an accidental emergent continuous rotationsymmetry.

By means of the effective theory for magnons and holes we derived the one-magnon ex-change potentials. Using these potentials, we investigated the Schrodinger equation describingthe relative motion of two holes in an otherwise undoped antiferromagnet. We solved the ra-dial part of this Schrodinger equation for a special case analytically and showed that undera condition on the low-energy constants the magnon-mediated forces lead to two hole boundstates.

We also investigated the weak coupling limit of the Hubbard model. In this contextwe derived the energy-momentum relation in the zero coupling limit, where massless Diracfermions arise. For these we constructed the leading terms of the corresponding effectiveLagrangian and derived an analytic expression for their velocity.

It would be interesting to solve the radial differential equations (9.11) numerically for thecases m 6= 0 in further studies and to see whether this leads to two-hole bound states. Ina further step, one could also take into account higher order potentials in the Schrodingerequation.

One knows that in real antiferromagnets, doped charge carriers get localized on impu-rities, which occur in every real antiferromagnet. Actually also doping, which is achievedby substituting ions in the insulating layers, causes impurities. This localization might be areason for the separation of the antiferromagnetic and the superconducting phases. It wouldalso be possible to include impurities in the effective theory, to investigate their influence on

59

60 10. Conclusions and Outlook

the low-energy physics of antiferromagnets. In this thesis we have laid the groundwork forsuch future investigations.

Acknowledgments

First I would like to thank Professor Uwe-Jens Wiese for his excellent support although hewas not in Bern all the time during this Master thesis. He always found time to answer a lotof questions, was understanding our difficulties and cared for a pleasant working atmosphere.I’m also grateful for his careful proof-reading.

Special thanks go to my mate Banz Bessire, with whom I did all work of his and myMaster thesis in close collaboration, for the excellent teamwork. It was a pleasure to workwith such a committed physicist.

Many thanks are addressed to Dr. Florian Kampfer who took, especially at the beginning,often time to answer our questions. This thanks also go to all the other people who helped usor we had discussions with, especially Dr. Fu-Jiung Jiang, Dr. Christoph Brugger, ProfessorChristoph Hofmann, and Urs Gerber.

Another big thanks is directed to my parents who made my studies possible although itwas my second education.

Last but not least, I thank Esther Fiechter, Ruth Bestgen, Ottilia Hanni, and KathrinWeyeneth for taking good care of all administrative issues.

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Bibliography

[1] F. Kampfer, Two-hole and two-electron bound states from a systematic low-energy effec-

tive field theory for magnons and charge carriers in an antiferromagnet, Doctoral Thesis,University of Bern (2007).

[2] F. Kampfer, Effective Field Theory for Charge Carriers in an Antiferromagnet, MasterThesis, University of Bern (2005).

[3] C. Brugger, Spiral Phases of Doped Antiferromagnets from a Systematic Low-Energy

Effective Field Theory, Doctoral Thesis, University of Bern (2008).

[4] M. R. Moser, Effective field theory for charge carriers in an antiferromagnet, DoctoralThesis, University of Bern (2006).

[5] C. Brugger, F. Kampfer, M. Moser, M. Pepe, and U.-J. Wiese, Phys. Rev. B74 (2006)224432.

[6] C. Brugger, F. Kampfer, M. Pepe, and U.-J. Wiese, Eur. Phys. J. B53 (2006) 433.

[7] F. Kampfer, M. Moser, and U.-J. Wiese, Nucl. Phys. B729 (2005) 317.

[8] C. Brugger, C. Hofmann, F. Kampfer, M. Moser, M. Pepe, and U.-J. Wiese, Phys. Rev.B75 (2007) 214405.

[9] B. Bessire, Spiral phases from a systematic low-energy effective field theory for magnons

and holes in an antiferromagnet on the honeycomb lattice, Master Thesis, University ofBern (2009).

[10] F.-J. Jiang, F. Kampfer, M. Nyfeler, and U.-J. Wiese, arXiv:0807.2977 [cond-mat.str-el](2008).

[11] F.-J. Jiang, F. Kampfer, C. P. Hofmann, and U.-J. Wiese, arXiv:0809.4998 [cond-mat.str-el] (2008).

[12] J. C. Bednorz and K. A. Muller, Z. Phys. B64 (1986) 189.

[13] J. Goldstone, Nuovo Cimento 19 (1961); J. Goldstone, A. Salam, and S. Weinberg, Phys.Rev. 127 (1962) 965.

[14] S. Weinberg, Physica 96 A (1979) 327.

[15] S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177 (1969) 2239.

63

64 Bibliography

[16] C. G. Callan, S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177 (1969) 2247.

[17] J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465.

[18] H. Georgi, Weak Interactions and Modern Particle Theory, Benjamin-Cummings Pub-lishing Company, 1984.

[19] J. Gasser, M. E. Sainio, and A. Svarc, Nucl. Phys. B307 (1988) 779.

[20] E. Jenkins and A. Manohar, Phys. Lett. B255 (1991) 558.

[21] V. Bernard, N. Kaiser, J. Kambor, and U.-G. Meissner, Nucl. Phys. B388 (1992) 315.

[22] T. Becher and H. Leutwyler, Eur. Phys. J. C9 (1999) 643.

[23] P. W. Anderson, Science 235 (1987) 1196.

[24] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. B39 (1989) 2344.

[25] U. Gerber, C. P. Hofmann, F.-J. Jiang, M. Nyfeler, U.-J. Wiese, arXiv:0901.0445 [cond-mat.str-el] (2009).

[26] C. N. Yang and S. Zhang, Mod. Phys. Lett. B4 (1990) 759.

[27] S. Zhang, Phys. Rev. Lett. 65 (1990) 120.

[28] P. Hasenfratz and H. Leutwyler, Nucl. Phys. B343 (1990) 241.

[29] P. Hasenfratz and F. Niedermayer, Z. Phys. B92 (1993) 91.

[30] P. Hasenfratz and F. Niedermayer, Phys. Lett. B268 (1991) 231.

[31] C. Brugger, C. Hofmann, F. Kampfer, M. Pepe, and U.-J. Wiese, Phys. Rev. B75 (2007)014421.

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

[33] S. Coleman, Commun. math. Phys. 31 (1973) 259.

[34] P. C. Hohenberg, Phys. Rev. 158 (1967) 383.

[35] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, arXiv:0709. 1163v2 [cond-mat. other](2008).

[36] P. R. Wallace, Phys. Rev. 71 (1947) 622.

[37] S. Sorella and E. Tosatti, Europhys. Lett 19, 699 (1992).

[38] I. F. Herbut Phys. Rev. Lett. 97, 146401 (2006).

[39] K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R. Dilanian, and T. Sasaki,Nature (London) 422 (2003) 53.

[40] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75 (2003) 473.

[41] R. Marshall, Proc. Roy. Soc. London A, 232 (1955) 48.

Bibliography 65

[42] N.W. Ashcroft, N.D. Mermin: Solid State Physics, Holt, Rinehart and Winston, 1976.

[43] E. Fradkin: Field Theories of condensed matter systems, Addison-Wesley publishingcompany, Redwood City, 1991.

[44] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publi-cations, Inc., New York.

Erklarung

gemass Art. 28 Abs. 2 RSL 05

Name/Vorname: Wirz Marcel

Matrikelnummer: 04-125-936

Studiengang: Physik

Bachelor 2 Master X2 Dissertation 2

Titel der Arbeit: Two-Hole Bound States from an EffectiveField Theory for Magnons and Holes in anAntiferromagnet on the Honeycomb Lattice

Leiter der Arbeit: Prof. Dr. Uwe-Jens Wiese

Ich erklare hiermit, dass ich diese Arbeit selbstandig verfasst und keine anderen als dieangegebenen Quellen benutzt habe. Alle Stellen, die wortlich oder sinngemass aus Quellenentnommen wurden, habe ich als solche gekennzeichnet. Mir ist bekannt, dass andernfalls derSenat gemass Artikel 36 Absatz 1 Buchstabe o des Gesetzes vom 5. September 1996 uber dieUniversitat zum Entzug des auf Grund dieser Arbeit verliehenen Titels berechtigt ist.

Bern, den 18. Februar 2009

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