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Published by

World Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication DataNames: Kuzemskiĭ, A. L. (Aleksandr Leonidovich), author.Title: Statistical mechanics and the physics of many-particle model systems / by Alexander Leonidovich Kuzemsky (Joint Institute for Nuclear Research, Russia).Description: Singapore ; Hackensack, NJ : World Scientific [2017] | Includes bibliographical references and index.Identifiers: LCCN 2016053884| ISBN 9789813145627 (hard cover ; alk. paper) | ISBN 9813145625 (hard cover ; alk. paper) | ISBN 9789813145634 (pbk ; alk. paper) | ISBN 9813145633 (pbk ; alk. paper)Subjects: LCSH: Quantum statistics. | Statistical mechanics. | Many-body problem-- Approximation methods. | Quantum theory. | Solid state physics.Classification: LCC QC174.4 .K84 2017 | DDC 530.13/3--dc23 LC record available at https://lccn.loc.gov/2016053884

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.

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KahFee - Statistical Mechanics and the Physics.indd 1 31-01-17 11:40:49 AM

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Dedicated to the memory of my teachersN. N. Bogoliubov and D. N. Zubarev

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Preface

We may say that equilibrium statistical mechanics is mainly statistical,whereas the nonequilibrium statistical mechanics is mainly mechanical.

Radu Balescu

“Statistical Mechanics and the Physics of Many-Particle Model Systems” isdevoted to the study of correlation effects in many-particle systems from aunified standpoint. The book is a self-contained and thorough treatment ofthe long-term researches that have been carried out in quantum-statisticalmechanics since 1969 by author and some other researchers belonging tothe N. N. Bogoliubov’s school. The book includes description of the funda-mental concepts and techniques of analysis, including recent developments.It also provides an overview that introduces main notions of the quantummany-particle physics with the emphasis on concepts and models. It treatsvarious actual systems having big significance in condensed matter physicsand quantum theory of magnetism. In addition, the book introduces basicconcepts and analytical methods for weakly interacting systems, and thenextends concepts and methods to strongly interacting systems. This bookcombines the features of textbook and research monograph and was writtenwith intention as a treatise. Most of this material cannot be found in anyother text. The book also contains an extensive bibliography.

The primary aim of this book is to provide a detailed account of a selectedgroup of results and developments in statistical mechanics and quantummany-particle physics in the approach of the N. N. Bogoliubov’s school. Theemphasis is on concepts and models and methods which are used in quantum-statistical physics. The fundamental works of N. N. Bogoliubov on statistical

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viii Preface

mechanics, many-body theory and quantum field theory [1–4], on the theoryof phase transitions, and on the general theory of interacting systems pro-vided a new perspective in various fields of mathematics and physics. Thepurpose of this book is to present the development of some advanced meth-ods of quantum statistical mechanics (equilibrium and nonequilibrium), andalso to show their effectiveness in applications to problems of quantum solid-state theory, and especially to problems of quantum theory of magnetism.The backbone of the entire book is the notion of thermodynamic two-time(Bogoliubov–Tyablikov) Green functions [5, 6].

Hence, the present text reflects the various techniques and developmentsin statistical mechanics in the approach of the N. N. Bogoliubov’s school,covering a variety of concepts and topics. The book is devoted to the inves-tigation of a series of problems of the strongly correlated electronic systemssuch as Hubbard, Anderson, spin–fermion models, model of disordered alloys,etc. A microscopic approach is developed to determine the quasiparticlesexcitation spectra of highly correlated electronic systems. For this purpose,the Green functions method is generalized to describe the excitation energet-ics in a self-consistent way within the irreducible Green functions method.The scheme was verified for a variety of systems and models of condensedmatter physics. Many topics presented here are usually not well covered instandard textbooks, therefore the aims of the present book were to pro-vide foundations and highlights on a variety of aspects in these fields withemphasis on the thorough discussion of the self-consistent approximationsfor suitable models.

The book contains introductory pedagogical chapters 1–14 with a hope tomake the presentation more coherent and self-contained. Many chapters alsoinclude varied additional information and discuss many complex researchareas which are not often discussed in other places.

Thus, those chapters 1–14 are intended as a brief summary and shortsurvey of the most important notions and concepts of quantum dynamicsand statistical mechanics for the sake of a self-contained formulation. Wetried to describe those concepts which have proven to be of value, and thosenotions which will be of use in clarifying subtle points.

The book is intended as a general course on the quantum many-bodyphysics at graduate and postgraduate levels for all-purpose physicists. Thetext is based mainly on my numerous lectures which were given in about50 Universities and Research Centers in nine countries. When preparing mylectures, I made numerous notes. It was designed to acquaint readers withkey concepts and their applications, to stimulate their own researches and togive in their hands a powerful and workable technique for doing that. Those

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Preface ix

notes (extended and rewritten in a coherent way) constituted the basis ofthe present book.

Thus, this text provides conceptual, technical, and factual guidanceon advanced topics of modern quantum solid-state theory and quantum-statistical physics. In addition, this book offers a unique and informativeoverview of various aspects of quantum many-particle physics and quantumtheory of condensed matter physics. The main part of the text is centeredaround the application of methods of the quantum field theory to study ofthe many-particle model systems, including the Green functions technique.We attempted to present both basic and more advanced topics of quan-tum many-particle physics in a mathematically consistent way, focusing onits operational ability as well as on physical consistency of the consideredmodels and methods.

It is necessary to stress that the path to understanding the founda-tions of the modern statistical mechanics and the development of efficientmethods for computing different physical characteristics of many-particlesystems was quite complex. The main postulates of the modern thermody-namics and statistical physics [7, 8] were formulated by J. P. Joule (1818–1889), R. Clausius (1822–1888), W. Thomson (1824–1907), J. C. Maxwell(1831–1879), L. Boltzmann (1844–1906), and, especially, by J. W. Gibbs(1839–1903). The foundational monograph by Gibbs “Elementary Principlesin Statistical Mechanics Developed with Special Reference to the RationalFoundations of Thermodynamics” [9–11] remains one of the highest peaksof modern theoretical science. A significant contribution to the developmentof modern methods of equilibrium and nonequilibrium statistical mechanicswas made by academician N. N. Bogoliubov (1909–1992) [1–4].

In the present book, selected short biographies of the key scientists areincluded at the end of some chapters to remind for readers (rather schemati-cally) the significant persons who contributed innovative ideas and methodsin the matter discussed.

It should be emphasized that specialists in theoretical physics, as well asexperimentalists, must be able to find their way through theoretical prob-lems of the modern physics of many-particle systems because of the followingreasons. Firstly, the statistical mechanics is filled with concepts, which widenthe physical horizon and general world outlook. Secondly, statistical mechan-ics and, especially, quantum statistical mechanics demonstrate remarkableefficiency and predictive ability achieved by constructing and applying fairlysimple (and at times even crude) many-particle models. Quite surprisingly,these simplified models allow one to describe a wide diversity of real sub-stances, materials, and the most nontrivial many-particle systems, such as

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x Preface

quark-gluon plasma, the DNA molecule, and interstellar matter. In systemsof many interacting particles, an important role is played by the so-calledcorrelation effects [12], which determine specific features in the behavior ofmost diverse objects, from cosmic systems to atomic nuclei. This is especiallytrue in the case of solid-state physics.

Indeed, the last few decades have witnessed an enormous develop-ment of condensed matter physics. A large number of data have accumu-lated and many experimental facts are known. Investigations of systemswith strong inter-electron correlations, complicated character of quasipar-ticle states, and strong potential scattering are extremely important andtopical problems of the modern theory of condensed matter. In addition,our time is marked by a rapid advancement in design and application ofnew materials, which not only find a wide range of applications in differ-ent areas of engineering, but they are also connected with the most fun-damental problems in physics, physical chemistry, molecular biology, andother branches of science. The quantum cooperative effects, such as mag-netism and superconductivity, frequently determine the unusual propertiesof these new materials. The same can also be said about other nontriv-ial quantum effects like, for instance, the quantum Hall effect, topologicalinsulators, Bose–Einstein condensation, quantum tunneling, and others. Thefurther development of our understanding of the new materials and com-plex substances and even some biological systems has depended, and stilldepend, on the development of more powerful experimental techniques formeasuring physical properties and more powerful theoretical techniques fordescribing and interpreting these properties. These research directions aredeveloping very rapidly, setting a fast pace for widening the domain wherethe methods of quantum-statistical mechanics are applied. The material ofthis book will support the above statements by concrete examples. Thebook also describes the resources and techniques necessary to realize thoseopportunities.

This book is devoted in its entirety to the following tasks. Its subjectmatter is theoretical condensed matter physics, by which we mean the the-oretical concepts, methods and models, and considerations which have beendevised in order to interpret the experimental material and to advance ourability to predict and control solid-state phenomena. Special attention waspaid to the unifying ideas and methods of the quantum-statistical mechan-ics. Obviously, this book does not pretend to cover all aspects of theoreticalcondensed matter physics. We were forced to omit many details and specialdevelopments. The omissions are due partly to the lack of space and partlyto the author’s personal preferences.

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It is worth mentioning that the author belongs to the N. N. Bogoliubov’sschool of theoretical physics. Thus, the methods formulated by N. N.Bogoliubov and his pupils were presented and elaborated in this book inthe most detailed way.

Chapters 15–29 are devoted to equilibrium quantum-statistical physics.The aim of Chapters 30–35 devoted to nonequilibrium statistical mechanicswas to provide better understanding of a few approaches that have beenproposed for treating nonequilibrium (time-dependent) processes in statisti-cal mechanics with the emphasis on the inter-relation between theories. Theensemble method, as it was formulated by J. W. Gibbs, has the great general-ity and the broad applicability to equilibrium statistical mechanics. Differentmacroscopic environmental constraints lead to different types of ensembleswith particular statistical characteristics. In the present book, the statisticaltheory of nonequilibrium processes, which is based on nonequilibrium ensem-ble formalism, is discussed. We also outlined the reasoning leading to someother useful approaches to the description of the irreversible processes. Thekinetic approach to dynamic many-body problems, which is important fromthe point of view of the fundamental theory of irreversibility, was alluded to.

The emphasis is on the method of the nonequilibrium statistical operator(NSO) developed by D. N. Zubarev [6]. The NSO method permits one togeneralize the Gibbs ensemble method to the nonequilibrium case and to con-struct an NSO which enables one to obtain the transport equations and cal-culate the transport coefficients in terms of correlation functions, and which,in the case of equilibrium, goes over to the Gibbs distribution. Although somespace was devoted to the formal structure of the NSO method, the emphasisis on its utility. Applications to specific problems such as the generalizedtransport and kinetic equations, and a few examples of the relaxation anddissipative processes, which manifest the operational ability of the method,are considered.

Hence, the book presents a broad selection of important topics, includingbasic models of many-particle interacting systems on a lattice and variousadvanced methods of equilibrium and nonequilibrium statistical mechanics.Throughout, emphasis has been placed on the logical structure of the theoryand consistent character of approximations. The course of study is aimed atunder-graduate, graduate, and post-graduate students and various researcheswho have had prior expose to the subject matter at a more elementary levelor have used other many-particle techniques. The part of the material isdiscussed at the level of an advanced course on quantum-statistical physics,highlighting a number of applications and useful examples, which in manycases goes beyond the material covered in many advanced undergraduate

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xii Preface

courses. The material of this book can also be of use in other contexts suchas mathematical physics and physics of materials.

Author profited greatly from fundamental insights offered in theRefs. [3–6]. Also, he found very useful many other texts and monographson statistical and many-body physics which he tried to cite in appropri-ate sections. However, author took care to avoid duplication of informationcovered in other books.

It is the hope of the author that the present book will serve as an intro-duction to the subject and a reference book and will help reader to appreci-ate vividly a beauty and elegance of statistical mechanics as an actual anddeveloping branch of contemporary science.

This book is dedicated to the memory of our common teacher academi-cian N. N. Bogoliubov (21.08.1909–13.02.1992). His inspiration, support andkind attention to author’s work are most gratefully acknowledged. I amgrateful to my teacher Prof. D. N. Zubarev (30.11.1917–16.07.1992). Thebook contains many marks of our numerous and sometimes hot discussionson statistical mechanics. I am also indebted for discussions and cooperationwith my friends Dr. L. A. Pokrovski, Prof. A. Holas, and Prof. K. Walasek(deceased). I am grateful to Prof. R. A. Minlos for the useful and clarifyingdiscussions on mathematical statistical mechanics topics. I also wish to thankmy collaborators for fruitful cooperation and to thank numerous other col-leagues of mine who in various ways contributed much to my understandingof the phenomena discussed.

A. L. KUZEMSKYBogoliubov Laboratory of Theoretical Physics,

Joint Institute for Nuclear Research,141980 Dubna, Moscow Region, Russia.

http://theor.jinr.ru/˜kuzemskye-mail: kuzemsky@theor.jinr.ru

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Contents

Preface vii

1. Probability, Information and Physics 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Theory of Probability . . . . . . . . . . . . . . . . . . . . . . 41.3 Logical Foundations of the Theory of Probability . . . . . . . 71.4 Principles of Statistical Description . . . . . . . . . . . . . . 91.5 Stochastic Processes and Probability . . . . . . . . . . . . . . 15

1.5.1 Normal or Gaussian Distribution . . . . . . . . . . . . 171.5.2 Poisson Distribution . . . . . . . . . . . . . . . . . . . 18

1.6 The Meaning of Probability . . . . . . . . . . . . . . . . . . . 191.7 Information and Probability . . . . . . . . . . . . . . . . . . . 221.8 Entropy and Information Theory . . . . . . . . . . . . . . . . 271.9 Biography of A. N. Kolmogorov . . . . . . . . . . . . . . . . 30

2. Dynamics of Particles 33

2.1 Classical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 332.2 Quantum Mechanics and Dynamics . . . . . . . . . . . . . . 38

2.2.1 States and Observables . . . . . . . . . . . . . . . . . 382.2.2 The Schrodinger Equation . . . . . . . . . . . . . . . 402.2.3 Expectation Values of Observables . . . . . . . . . . . 422.2.4 Probability and Normalization

of the Wave Functions . . . . . . . . . . . . . . . . . . 452.2.5 Gram–Schmidt Orthogonalization Procedure . . . . . 49

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2.3 Evolution of Quantum System . . . . . . . . . . . . . . . . . 512.3.1 Time Evolution and Stationary States . . . . . . . . . 512.3.2 Dynamical Behavior of Quantum System . . . . . . . 53

2.4 The Density Operator . . . . . . . . . . . . . . . . . . . . . . 592.4.1 The Change in Time of the Density Matrix . . . . . . 632.4.2 The Ehrenfest Theorem . . . . . . . . . . . . . . . . . 65

2.5 Biography of Erwin Schrodinger . . . . . . . . . . . . . . . . 67

3. Perturbation Theory 71

3.1 Perturbation Techniques . . . . . . . . . . . . . . . . . . . . . 713.2 Rayleigh–Schrodinger Perturbation Theory . . . . . . . . . . 733.3 The Brillouin–Wigner Perturbation Theory . . . . . . . . . . 763.4 Comparison of Rayleigh–Schrodinger and Brillouin–Wigner

Perturbation Theories . . . . . . . . . . . . . . . . . . . . . . 813.5 The Variational Principles of Quantum Theory . . . . . . . . 853.6 Time-Dependent Perturbation Theory . . . . . . . . . . . . . 873.7 Transition Rate and Fermi Golden Rule . . . . . . . . . . . . 903.8 Specific Perturbations and Transition Rate . . . . . . . . . . 933.9 The Natural Width of a Spectral Line . . . . . . . . . . . . . 963.10 Decay Rates for Quantum Systems . . . . . . . . . . . . . . . 993.11 Effect of Relaxation on a Resonance Absorption

Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.12 Transition Probability due to Random Perturbations . . . . . 104

4. Scattering Theory 107

4.1 Scattering Problem: General Approach . . . . . . . . . . . . 1074.2 Potential Scattering . . . . . . . . . . . . . . . . . . . . . . . 1094.3 Scattering Cross-Section . . . . . . . . . . . . . . . . . . . . . 1104.4 Scattering Theory and the Transition Rate . . . . . . . . . . 1124.5 The Formal Scattering Theory . . . . . . . . . . . . . . . . . 117

5. Green Functions Method in Mathematical Physics 121

5.1 The Green Functions and the Differential Equations . . . . . 1215.2 The Green Function of the Schrodinger Equation . . . . . . . 1265.3 The Propagation of a Wave Function . . . . . . . . . . . . . 1335.4 Time-dependent Green Functions

and Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . 137

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Contents xv

5.5 The Energy-Dependent Green Functions . . . . . . . . . . . . 1425.6 The Green Functions and the Scattering Problem . . . . . . 1465.7 Principles of Limiting Absorption and Limiting Amplitude

in Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . 1495.8 Biography of George Green . . . . . . . . . . . . . . . . . . . 152

6. Symmetry and Invariance 155

6.1 Symmetry Principles in Physics . . . . . . . . . . . . . . . . . 1556.2 Groups and Symmetry Transformations . . . . . . . . . . . . 1586.3 Symmetry in Quantum Mechanics . . . . . . . . . . . . . . . 1616.4 Invariance and Conservation Properties . . . . . . . . . . . . 1646.5 The Physics of Time Reversal . . . . . . . . . . . . . . . . . . 1676.6 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 1696.7 Biography of Pierre Curie . . . . . . . . . . . . . . . . . . . . 173

7. The Angular Momentum and Spin 177

7.1 Space-Rotation Invariance . . . . . . . . . . . . . . . . . . . . 1777.2 Angular Momentum Operator . . . . . . . . . . . . . . . . . 1817.3 The Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1847.4 Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . 1897.5 Exchange Forces and Microscopic Origin

of Spin Interactions . . . . . . . . . . . . . . . . . . . . . . . 1927.6 Time Reversal Symmetry . . . . . . . . . . . . . . . . . . . . 197

8. Equilibrium Statistical Thermodynamics 201

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.2 Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . 2058.3 Gibbs Ensembles Method . . . . . . . . . . . . . . . . . . . . 2138.4 Gibbs and Boltzmann Entropy . . . . . . . . . . . . . . . . . 2208.5 The Canonical Distribution and Gibbs Theorem . . . . . . . 2238.6 Ensembles in Quantum-Statistical Mechanics . . . . . . . . . 2268.7 Biography of J. W. Gibbs . . . . . . . . . . . . . . . . . . . . 231

9. Dynamics and Statistical Mechanics 235

9.1 Interrelation of Dynamics and Statistical Mechanics . . . . . 2359.2 Equipartition of Energy . . . . . . . . . . . . . . . . . . . . . 2419.3 Nonlinear Oscillations and Time Averaging . . . . . . . . . . 244

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10. Thermodynamic Limit in Statistical Mechanics 253

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 25310.2 Thermodynamic Limit in Statistical Physics . . . . . . . . . 25610.3 Equivalence and Nonequivalence of Ensembles . . . . . . . . 26210.4 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . 26610.5 Small and Non-Standard Systems . . . . . . . . . . . . . . . 27110.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 272

11. Maximum Entropy Principle 275

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 27511.2 Maximum Entropy Principle . . . . . . . . . . . . . . . . . . 27611.3 Applicability of the Maximum Entropy Algorithm . . . . . . 28711.4 Biography of E. T. Jaynes . . . . . . . . . . . . . . . . . . . . 293

12. Band Theory and Electronic Properties of Solids 297

12.1 Solid State: Metals and Nonmetals . . . . . . . . . . . . . . . 29712.2 Energy Band Structure of Metals and Nonmetals . . . . . . . 29812.3 Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 30312.4 Atomic Orbitals and Tight-Binding Approximation . . . . . . 306

12.4.1 Localized atomic and molecular orbitals . . . . . . . . 30712.4.2 Tight-binding approximation . . . . . . . . . . . . . . 309

12.5 Effective Electron Mass . . . . . . . . . . . . . . . . . . . . . 31312.6 Biography of Felix Bloch . . . . . . . . . . . . . . . . . . . . 316

13. Magnetic Properties of Substances and Materials 319

13.1 Solid-State Physics of Complex Materials . . . . . . . . . . . 31913.2 Physics of Magnetism . . . . . . . . . . . . . . . . . . . . . . 32113.3 Quantum Theory of Magnetism . . . . . . . . . . . . . . . . . 32713.4 Localized Models of Magnetism . . . . . . . . . . . . . . . . . 329

13.4.1 Ising model . . . . . . . . . . . . . . . . . . . . . . . . 32913.4.2 Heisenberg model . . . . . . . . . . . . . . . . . . . . 330

13.5 Problem of Magnetism of Itinerant Electrons . . . . . . . . . 33413.6 Biography of Pierre-Ernest Weiss . . . . . . . . . . . . . . . . 336

14. Statistical Physics of Many-Particle Systems 339

14.1 Theory of Many-Particle Systems with Interactions . . . . . . 33914.2 The Method of Second Quantization . . . . . . . . . . . . . . 34114.3 Chemical Potential of Many-Particle Systems . . . . . . . . . 348

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14.4 Model of Dilute Bose Gas . . . . . . . . . . . . . . . . . . . . 35114.5 Bogoliubov Canonical Transformation Method . . . . . . . . 35614.6 Model Description of Complex Materials . . . . . . . . . . . 36014.7 Itinerant Electron Models . . . . . . . . . . . . . . . . . . . . 36214.8 Interacting Electrons on a Lattice

and the Hubbard Model . . . . . . . . . . . . . . . . . . . . . 36414.9 The Anderson Model . . . . . . . . . . . . . . . . . . . . . . 36814.10 Multi-Band Models. Model with s–d Hybridization . . . . . . 36914.11 The s–d Exchange Model and the Zener Model . . . . . . . . 36914.12 Falicov–Kimball Model . . . . . . . . . . . . . . . . . . . . . 37014.13 Model of Disordered Binary Substitutional Alloys . . . . . . 37114.14 The Adequacy of the Model Description . . . . . . . . . . . . 375

15. Thermodynamic Green Functions 379

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 37915.2 Methods of the Quantum Field Theory and Many-Particle

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38015.3 Variety of the Green Functions . . . . . . . . . . . . . . . . . 38115.4 Temperature Green Functions . . . . . . . . . . . . . . . . . 38615.5 Two-Time Temperature Green Functions . . . . . . . . . . . 38915.6 Green Functions and Time Correlation Functions . . . . . . . 39715.7 Quasiparticle Many-Body Dynamics . . . . . . . . . . . . . . 40315.8 The Method of Irreducible Green Functions . . . . . . . . . . 40715.9 Green Functions and Moments of Spectral Density . . . . . . 41215.10 Projection Methods and the Irreducible

Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . 41615.11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 41715.12 Biography of J. Schwinger . . . . . . . . . . . . . . . . . . . . 418

16. Applications of the Green Functions Method 421

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 42116.2 Perfect Quantum Gases . . . . . . . . . . . . . . . . . . . . . 42216.3 Green Functions and Perturbation Theory . . . . . . . . . . . 42316.4 Natural Width of a Spectral Line and Green Functions . . . 42816.5 Scattering of Neutrons by Condensed Matter . . . . . . . . . 433

16.5.1 The Transition Rate . . . . . . . . . . . . . . . . . . . 43416.5.2 Transition Amplitude and Cross-section . . . . . . . . 43516.5.3 Scattering Function and Cross-section . . . . . . . . . 440

16.6 Biography of Dirk ter Haar . . . . . . . . . . . . . . . . . . . 443

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17. Spin Systems and the Green Functions Method 447

17.1 Spin Systems on a Lattice . . . . . . . . . . . . . . . . . . . . 44717.2 Heisenberg Antiferromagnet . . . . . . . . . . . . . . . . . . . 44817.3 Green Functions and Isotropic Heisenberg Model . . . . . . . 45017.4 Heisenberg Ferromagnet and Boson Representation . . . . . . 45417.5 Tyablikov and Callen Decoupling . . . . . . . . . . . . . . . . 45817.6 Rotational Invariance and Heisenberg Model . . . . . . . . . 46017.7 Heisenberg Model of Spin System with Two Spins

Per Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46217.8 Spin-Wave Scattering Effects in Heisenberg

Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . 47117.9 Heisenberg Antiferromagnet at Finite Temperatures . . . . . 475

17.9.1 Hamiltonian of the Model . . . . . . . . . . . . . . . . 47517.9.2 Quasiparticle Many-Body Dynamics of Heisenberg

Antiferromagnet . . . . . . . . . . . . . . . . . . . . . 47617.9.3 GMF Green Function . . . . . . . . . . . . . . . . . . 47917.9.4 Damping of Quasiparticle Excitations . . . . . . . . . 481

17.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48317.11 Biography of S. V. Tyablikov . . . . . . . . . . . . . . . . . . 484

18. Correlated Fermion Systems on a Lattice:Hubbard Model 487

18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 48718.2 Quasiparticle Many-Body Dynamics of Lattice

Fermion Models . . . . . . . . . . . . . . . . . . . . . . . . . 49018.3 Hubbard Model: Weak Correlation . . . . . . . . . . . . . . . 49418.4 Hubbard Model: Strong Correlation . . . . . . . . . . . . . . 50118.5 Correlations in Random Hubbard Model . . . . . . . . . . . . 50918.6 Interpolation Solutions of Correlated Models . . . . . . . . . 51418.7 Effective Perturbation Expansion for the Self-Energy

Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51518.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51618.9 Biography of John Hubbard . . . . . . . . . . . . . . . . . . . 518

19. Correlated Fermion Systems on a Lattice.Anderson Model 521

19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 52119.2 Hamiltonian of the Models . . . . . . . . . . . . . . . . . . . 523

19.2.1 Single-impurity Anderson model . . . . . . . . . . . . 523

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19.2.2 Periodic Anderson model . . . . . . . . . . . . . . . . 52419.2.3 Two-Impurity Anderson model . . . . . . . . . . . . . 525

19.3 The Irreducible Green Functions Method and SIAM . . . . . 52519.4 SIAM: Strong Correlation . . . . . . . . . . . . . . . . . . . . 52819.5 IGF Method and Interpolation Solution of SIAM . . . . . . . 53019.6 Quasiparticle Dynamics of SIAM . . . . . . . . . . . . . . . . 53319.7 Complex Expansion for a Propagator . . . . . . . . . . . . . 53519.8 The Improved Interpolative Treatment of SIAM . . . . . . . 53919.9 Quasiparticle Many-Body Dynamics of PAM . . . . . . . . . 54419.10 Quasiparticle Many-Body Dynamics of TIAM . . . . . . . . . 54619.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

20. Spin–Fermion Model of Magnetism: QuasiparticleMany-Body Dynamics 553

20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 55320.2 The Spin-Fermion Model . . . . . . . . . . . . . . . . . . . . 55520.3 Quasiparticle Dynamics of the (sp–d) Model . . . . . . . . . 55720.4 Spin Dynamics of the sp–d Model: Scattering Regime . . . . 55920.5 Generalized Mean-Field Green Function . . . . . . . . . . . . 56520.6 Uncoupled Subsystems . . . . . . . . . . . . . . . . . . . . . . 56520.7 Coupled Subsystems . . . . . . . . . . . . . . . . . . . . . . . 56820.8 Effects of Disorder in Diluted Magnetic Semiconductors . . . 57220.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

21. Spin–Fermion Model of Magnetism: Theoryof Magnetic Polaron 577

21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 57721.2 Charge and Spin Degrees of Freedom . . . . . . . . . . . . . 58121.3 Charge Dynamics of the s–d Model: Scattering Regime . . . 58821.4 Charge Dynamics of the s–d Model: Bound

State Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 59021.5 Magnetic Polaron in GMF . . . . . . . . . . . . . . . . . . . 59721.6 Damping of the Magnetic Polaron State . . . . . . . . . . . . 60021.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 606

22. Quantum Protectorate and Microscopic Modelsof Magnetism 609

22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 60922.2 Magnetic Degrees of Freedom . . . . . . . . . . . . . . . . . . 610

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22.3 Microscopic Picture of Magnetism in Materials . . . . . . . . 61122.3.1 Heisenberg model . . . . . . . . . . . . . . . . . . . . 61222.3.2 Itinerant electron model . . . . . . . . . . . . . . . . . 61222.3.3 Hubbard model . . . . . . . . . . . . . . . . . . . . . 61322.3.4 Multi-band model: model with s–d

hybridization . . . . . . . . . . . . . . . . . . . . . . . 61422.3.5 Spin–fermion model . . . . . . . . . . . . . . . . . . . 614

22.4 Symmetry and Physics of Magnetism . . . . . . . . . . . . . 61522.5 Spin Quasiparticle Dynamics . . . . . . . . . . . . . . . . . . 617

22.5.1 Spin dynamics of the Hubbard model . . . . . . . . . 61822.5.2 Spin dynamics of the SFM . . . . . . . . . . . . . . . 61822.5.3 Spin dynamics of the multi-band model . . . . . . . . 621

22.6 Quasiparticle Excitation Spectra and NeutronScattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

22.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

23. Quasiaverages and Symmetry Breaking 631

23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 63123.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 63423.3 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . 63723.4 Goldstone Theorem . . . . . . . . . . . . . . . . . . . . . . . 64323.5 Higgs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 64623.6 Bogoliubov Quasiaverages in Statistical Mechanics . . . . . . 649

23.6.1 Bogoliubov theorem on the singularity of 1/q2 . . . . 65923.6.2 Bogoliubov’s inequality and the Mermin–Wagner

theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 66423.7 Broken Symmetries and Condensed Matter Physics . . . . . 676

23.7.1 Superconductivity . . . . . . . . . . . . . . . . . . . . 67823.7.2 Antiferromagnetism . . . . . . . . . . . . . . . . . . . 68523.7.3 Bose systems . . . . . . . . . . . . . . . . . . . . . . . 693

23.8 Conclusions and Discussions . . . . . . . . . . . . . . . . . . 69623.9 Biography of N. N. Bogoliubov . . . . . . . . . . . . . . . . . 697

24. Emergence and Emergent Phenomena 703

24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 70324.2 Emergence Concept . . . . . . . . . . . . . . . . . . . . . . . 70524.3 Emergent Phenomena . . . . . . . . . . . . . . . . . . . . . . 70624.4 Quantum Mechanics and its Emergent Macrophysics . . . . . 709

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24.5 Emergent Phenomena in Quantum CondensedMatter Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 713

24.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

25. Electron–Lattice Interaction in Metals and Alloys 719

25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 71925.2 Electron-Lattice Interaction in Condensed

Matter Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 72125.3 Modified Tight-Binding Approximation . . . . . . . . . . . . 726

25.3.1 Electron Green function . . . . . . . . . . . . . . . . . 73125.3.2 Phonon Green function . . . . . . . . . . . . . . . . . 73325.3.3 Renormalization of the electron

and phonon spectra . . . . . . . . . . . . . . . . . . . 73425.4 Renormalized Spectrum of the Mott–Hubbard Insulator . . . 73625.5 The Electron–Phonon Spectral Function . . . . . . . . . . . . 73925.6 Electron–Lattice Interaction in Disordered Binary Alloys . . 744

25.6.1 The model . . . . . . . . . . . . . . . . . . . . . . . . 74525.6.2 Electron Green functions for alloy . . . . . . . . . . . 74825.6.3 Green function for lattice vibrations in alloy . . . . . 75125.6.4 The configurational averaging . . . . . . . . . . . . . 75225.6.5 The electronic specific heat . . . . . . . . . . . . . . . 756

25.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75725.8 Biography of Herbert Frohlich . . . . . . . . . . . . . . . . . 759

26. Superconductivity in Transition Metalsand their Disordered Alloys 763

26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 76326.2 The Microscopic Theory of Superconductivity . . . . . . . . . 765

26.2.1 The Nambu formalism . . . . . . . . . . . . . . . . . 76726.2.2 The Eliashberg equations . . . . . . . . . . . . . . . . 768

26.3 Equations of Superconductivity in WannierRepresentations . . . . . . . . . . . . . . . . . . . . . . . . . 775

26.4 Strong-Coupling Equations of Superconductivity . . . . . . . 78026.5 Equations of Superconductivity in Disordered

Binary Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . 78326.5.1 The model Hamiltonian . . . . . . . . . . . . . . . . . 78426.5.2 Electron Green function and mass operator . . . . . . 78526.5.3 Renormalized lattice Green function . . . . . . . . . . 78926.5.4 Configurational averaging . . . . . . . . . . . . . . . . 789

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26.5.5 Simplest method of averaging . . . . . . . . . . . . . 79026.5.6 General averaging scheme . . . . . . . . . . . . . . . . 79226.5.7 The random contact model . . . . . . . . . . . . . . . 79226.5.8 CPA equations for superconductivity

in the contact model . . . . . . . . . . . . . . . . . . 79426.5.9 Linearized equations and transition temperature . . . 795

26.6 The Physics of Layered Systems and Superconductivity . . . 79826.6.1 Layered cuprates . . . . . . . . . . . . . . . . . . . . . 80026.6.2 Crystal structure of cuprates

and mercurocuprates . . . . . . . . . . . . . . . . . . 80326.6.3 The Lawrence–Doniach model . . . . . . . . . . . . . 805

26.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809

27. Spectral Properties of the GeneralizedSpin-Fermion Models 811

27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 81127.2 Generalized SFM . . . . . . . . . . . . . . . . . . . . . . . . . 81227.3 Spin Dynamics of the d–f Model . . . . . . . . . . . . . . . . 813

27.3.1 Generalized Mean-field Green Function . . . . . . . . 81727.3.2 Dyson Equation for d–f Model . . . . . . . . . . . . . 819

27.4 Self-Energy and Damping . . . . . . . . . . . . . . . . . . . . 82227.5 Charge Dynamics of the d–f Model . . . . . . . . . . . . . . 82427.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828

28. Correlation Effects in High-Tc Superconductorsand Heavy Fermion Compounds 829

28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 82928.2 The Electronic Properties of Correlated Systems . . . . . . . 83028.3 The Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . 83228.4 The Effective Hamiltonians . . . . . . . . . . . . . . . . . . . 83428.5 Coexistence of Spin and Carrier Subsystems . . . . . . . . . 83728.6 Competition of Interactions in Kondo Systems . . . . . . . . 84028.7 Dynamics of Carriers in the Spin–Fermion Model . . . . . . . 842

28.7.1 Hole dynamics in cuprates . . . . . . . . . . . . . . . 84228.7.2 Hubbard model and t–J model . . . . . . . . . . . . . 84328.7.3 Hole spectrum of t–J model . . . . . . . . . . . . . . 84428.7.4 The Kondo–Heisenberg model . . . . . . . . . . . . . 84628.7.5 Hole dynamics in the Kondo–Heisenberg model . . . 84728.7.6 Dynamics of spin subsystem . . . . . . . . . . . . . . 850

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28.7.7 Damping of hole quasiparticles . . . . . . . . . . . . . 85228.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 855

29. Generalized Mean Fields and VariationalPrinciple of Bogoliubov 857

29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 85729.2 The Helmholtz Free Energy . . . . . . . . . . . . . . . . . . . 85829.3 Approximate Calculations of the Helmholtz Free Energy . . . 85929.4 The Mean Field Concept . . . . . . . . . . . . . . . . . . . . 86429.5 The Mathematical Tools . . . . . . . . . . . . . . . . . . . . . 87029.6 Variational Principle of Bogoliubov . . . . . . . . . . . . . . . 87529.7 Applications of the Bogoliubov Variational Principle . . . . . 88129.8 The Variational Schemes and Bounds on Free Energy . . . . 88529.9 The Hartree–Fock–Bogoliubov Mean Fields . . . . . . . . . . 88929.10 Method of an Approximating Hamiltonian . . . . . . . . . . 89729.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900

30. Nonequilibrium Statistical Thermodynamics 903

30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 90330.2 Ensemble Method in the Theory of Irreversibility . . . . . . . 90530.3 Statistical Mechanics of Irreversibility . . . . . . . . . . . . . 90830.4 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . 91130.5 The Method of Time Correlation Functions . . . . . . . . . . 91730.6 Kubo Linear Response Theory . . . . . . . . . . . . . . . . . 91830.7 Fluctuation Theorem and Green–Kubo Relations . . . . . . . 92030.8 Conditions for Local Equilibrium . . . . . . . . . . . . . . . . 92330.9 Modified Projection Methods . . . . . . . . . . . . . . . . . . 928

31. Method of the Nonequilibrium Statistical Operator 931

31.1 Nonequilibrium Ensembles . . . . . . . . . . . . . . . . . . . 93131.2 The NSO Method . . . . . . . . . . . . . . . . . . . . . . . . 93331.3 The Relevant Operators . . . . . . . . . . . . . . . . . . . . . 93431.4 Construction of the NSO . . . . . . . . . . . . . . . . . . . . 935

32. Nonequilibrium Statistical Operatorand Transport Equations 945

32.1 Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . . 94532.2 The Transport and Kinetic Equations . . . . . . . . . . . . . 947

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32.3 System in Thermal Bath: GeneralizedKinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . 949

32.4 System in Thermal Bath: Rate and Master Equations . . . . 95232.5 A Dynamical System in a Thermal Bath . . . . . . . . . . . . 95432.6 Schrodinger-type Equation with Damping for a Dynamical

System in a Thermal Bath . . . . . . . . . . . . . . . . . . . 960

33. Applications of the NonequilibriumStatistical Operator 965

33.1 Damping Effects in a System Interactingwith a Thermal Bath . . . . . . . . . . . . . . . . . . . . . . 965

33.2 The Natural Width of Spectral Lineof the Atomic System . . . . . . . . . . . . . . . . . . . . . . 967

33.3 Evolution of a System in an Alternating External Field . . . 97033.4 Statistical Theory of Spin Relaxation and Diffusion

in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97933.4.1 Dynamics of nuclear spin system . . . . . . . . . . . . 98233.4.2 Nuclear spin–lattice relaxation . . . . . . . . . . . . . 98533.4.3 Spin diffusion of nuclear magnetic moment . . . . . . 98833.4.4 Spin diffusion coefficient . . . . . . . . . . . . . . . . 99033.4.5 Stochasticity of spin subsystem . . . . . . . . . . . . . 99433.4.6 Spin diffusion coefficient in dilute alloys . . . . . . . . 997

33.5 Other Applications of the NSO Method . . . . . . . . . . . . 100133.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003

34. Generalized Van Hove Formula for Scatteringof Particles by Statistical Medium 1005

34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 100534.2 Density Correlation Function . . . . . . . . . . . . . . . . . . 100834.3 Scattering Function and Cross-Section . . . . . . . . . . . . . 100934.4 The Van Hove Formula . . . . . . . . . . . . . . . . . . . . . 101134.5 Van Hove Formalism for the Nonequilibrium Statistical

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101634.6 Scattering of Beam of Particles by the Nonequilibrium

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101734.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 102334.8 Biography of Leon Van Hove . . . . . . . . . . . . . . . . . . 1024

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35. Electronic Transport in Metallic Systems 1029

35.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 102935.2 Many-Particle Interacting Systems and Current

Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103335.3 Current Operator for the Tight-Binding Electrons . . . . . . 103735.4 Charge and Heat Transport . . . . . . . . . . . . . . . . . . . 1040

35.4.1 Electrical resistivity and Ohm’s law . . . . . . . . . . 104035.4.2 Drude–Lorentz model . . . . . . . . . . . . . . . . . 1041

35.5 The Temperature Dependence of Conductivity . . . . . . . . 104535.6 Conductivity of Alloys . . . . . . . . . . . . . . . . . . . . . . 105135.7 Magnetoresistance and the Hall Effect . . . . . . . . . . . . . 105435.8 Thermal Conduction in Solids . . . . . . . . . . . . . . . . . 105635.9 Linear Macroscopic Transport Equations . . . . . . . . . . . 105835.10 Statistical Mechanics and Transport Coefficients . . . . . . . 106135.11 Transport Theory and Electrical Conductivity . . . . . . . . 106235.12 Method of Time Correlation Functions and Linear

Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . 106435.13 Green Functions in the Theory of Irreversible Processes . . . 107135.14 The Electrical Conductivity Tensor . . . . . . . . . . . . . . 107635.15 Linear Response Theory: Pro et Contra . . . . . . . . . . . . 107835.16 Generalized Kinetic Equations . . . . . . . . . . . . . . . . . 108635.17 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . 108835.18 Resistivity of Transition Metal with NonSpherical Fermi

Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109135.19 Temperature Dependence of Resistivity . . . . . . . . . . . . 109635.20 Equivalence of NSO Approach and Kubo Formalism . . . . . 109935.21 High-Temperature Resistivity and MTBA . . . . . . . . . . . 110235.22 Resistivity of Disordered Alloys . . . . . . . . . . . . . . . . . 111335.23 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112435.24 Biography of Georg Simon Ohm . . . . . . . . . . . . . . . . 1126

Bibliography 1129

Index 1225 Sta

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