Statistical Mechanics and the Physics of Many-Particle Model Systems

The following sections are included:

• Dedication
• Preface
• Contents

First, a quick survey of some background material will be useful. Hence, Chapters 1–14 can be considered as an extended introduction in the core content of the book. It is intended as a brief summary and short survey of the most important notions and concepts of dynamics and statistical mechanics for the sake of a self-contained formulation. We wish to describe those concepts which have proven to be of value, and those notions which will be of use in clarifying subtle points.

The following sections are included:

• Classical Dynamics
• Quantum Mechanics and Dynamics
  • States and Observables
  • The Schrödinger Equation
  • Expectation Values of Observables
  • Probability and Normalization of the Wave Functions
  • Gram–Schmidt Orthogonalization Procedure
• Evolution of Quantum System
  • Time Evolution and Stationary States
  • Dynamical Behavior of Quantum System
• The Density Operator
  • The Change in Time of the Density Matrix
  • The Ehrenfest Theorem
• Biography of Erwin Schrödinger

The following sections are included:

• Perturbation Techniques
• Rayleigh–Schrödinger Perturbation Theory
• The Brillouin–Wigner Perturbation Theory
• Comparison of Rayleigh–Schrödinger and Brillouin–Wigner Perturbation Theories
• The Variational Principles of Quantum Theory
• Time-Dependent Perturbation Theory
• Transition Rate and Fermi Golden Rule
• Specific Perturbations and Transition Rate
• The Natural Width of a Spectral Line
• Decay Rates for Quantum Systems
• Effect of Relaxation on a Resonance Absorption Spectrum
• Transition Probability due to Random Perturbations

The following sections are included:

• Scattering Problem: General Approach
• Potential Scattering
• Scattering Cross-Section
• Scattering Theory and the Transition Rate
• The Formal Scattering Theory

The Green functions technique is a method to solve a nonhomogeneous differential equation. The essence of the method consists in finding an integral operator which produces a solution satisfying all given boundary conditions. The Green function is the kernel of the integral operator inverse to the differential operator generated by the given differential equation and the homogeneous boundary conditions. It reduces the study of the properties of the differential operator to the study of similar properties of the corresponding integral operator. The integral operator has a kernel called the Green function, usually denoted G(x, t). This is multiplied by the nonhomogeneous term and integrated by one of the variables.

This chapter introduces briefly some important notions of theoretical physics such as symmetry and invariance. In particular, space-translation, spacerotation, space-inversion, Galilean, and time-invariance are discussed tersely. The discussion of this chapter is rather heuristic and is meant to serve as an introduction to the basic notions and techniques associated with symmetry and invariance principles. Although the discussion of this material is mainly qualitative, all the necessary references for a deeper study were given. Additional detailed discussion on these notions will be provided further in this book in relation with the concrete problems.

In the present chapter, we remind briefly the important notions of the angular momentum and spin. Angular momentum has direction and magnitude and is a conserved quantity. In a suitable basis, one can expand the angular momentum in terms of components. Conservation of angular momentum implies that each component is separately conserved.

The following sections are included:

• Introduction
• Statistical Thermodynamics
• Gibbs Ensembles Method
• Gibbs and Boltzmann Entropy
• The Canonical Distribution and Gibbs Theorem
• Ensembles in Quantum-Statistical Mechanics
• Biography of J.W. Gibbs

The following sections are included:

• Interrelation of Dynamics and Statistical Mechanics
• Equipartition of Energy
• Nonlinear Oscillations and Time Averaging

The thermodynamic limit in statistical thermodynamics of many-particle systems is an important but often overlooked issue in the various applied studies of condensed matter physics. To clarify this issue, we will discuss tersely the past and present disposition of thermodynamic limiting procedure in the structure of the contemporary statistical mechanics and our current understanding of this problem [467]. We pick out the ingenious approach by N. N. Bogoliubov, who developed a general formalism for establishing the limiting distribution functions in the form of formal series in powers of the density. In this study, he outlined the method of justification of the thermodynamic limit when he derived the generalized Boltzmann equations. We take this opportunity to give a brief survey of the closely related problem of the equivalence and nonequivalence of statistical ensembles. The major aim of this chapter is to provide a better qualitative understanding of the physical significance of the thermodynamic limit in modern statistical physics of the infinite and “small” many-particle systems.

The following sections are included:

• Introduction
• Maximum Entropy Principle
• Applicability of the Maximum Entropy Algorithm
• Biography of E. T. Jaynes

The science of solid state [687] plays an important role in various applications in modern technology and industry. The solid-state physics needed the methods of quantum theory and statistical mechanics. In various applied problems, the quantum theory and statistical mechanics offer a useful and workable language of a great predictive power for describing the basic properties of solids and for predicting the new functional materials. Solid-state physics is related tightly with the major overlapping research field within solid-state science [687, 688]. The basic electronic properties of materials provide a basis for a useful classification according to the nature of electron states in the material.

The following sections are included:

• Solid-State Physics of Complex Materials
• Physics of Magnetism
• Quantum Theory of Magnetism
• Localized Models of Magnetism
  • Ising model
  • Heisenberg model
• Problem of Magnetism of Itinerant Electrons
• Biography of Pierre-Ernest Weiss

The following sections are included:

• Theory of Many-Particle Systems with Interactions
• The Method of Second Quantization
• Chemical Potential of Many-Particle Systems
• Model of Dilute Bose Gas
• Bogoliubov Canonical Transformation Method
• Model Description of Complex Materials
• Itinerant Electron Models
• Interacting Electrons on a Lattice and the Hubbard Model
• The Anderson Model
• Multi-Band Models. Model with s–d Hybridization
• The s–d Exchange Model and the Zener Model
• Falicov–Kimball Model
• Model of Disordered Binary Substitutional Alloys
• The Adequacy of the Model Description

The following sections are included:

• Introduction
• Methods of the Quantum Field Theory and Many-Particle Systems
• Variety of the Green Functions
• Temperature Green Functions
• Two-Time Temperature Green Functions
• Green Functions and Time Correlation Functions
• Quasiparticle Many-Body Dynamics
• The Method of Irreducible Green Functions
• Green Functions and Moments of Spectral Density
• Projection Methods and the Irreducible Green Functions
• Concluding Remarks
• Biography of J. Schwinger

In the present chapter, we shall discuss a few typical applications of the double-time temperature dependent Green functions to specific problems.

The following sections are included:

• Spin Systems on a Lattice
• Heisenberg Antiferromagnet
• Green Functions and Isotropic Heisenberg Model
• Heisenberg Ferromagnet and Boson Representation
• Tyablikov and Callen Decoupling
• Rotational Invariance and Heisenberg Model
• Heisenberg Model of Spin System with Two Spins Per Site
• Spin-Wave Scattering Effects in Heisenberg Ferromagnet
• Heisenberg Antiferromagnet at Finite Temperatures
  • Hamiltonian of the Model
  • Quasiparticle Many-Body Dynamics of Heisenberg Antiferromagnet
  • GMF Green Function
  • Damping of Quasiparticle Excitations
• Conclusions
• Biography of S. V. Tyablikov

The following sections are included:

• Introduction
• Quasiparticle Many-Body Dynamics of Lattice Fermion Models
• Hubbard Model: Weak Correlation
• Hubbard Model: Strong Correlation
• Correlations in Random Hubbard Model
• Interpolation Solutions of Correlated Models
• Effective Perturbation Expansion for the Self-Energy Operator
• Conclusions
• Biography of John Hubbard

In this chapter, we discuss the many-body quasiparticle dynamics of the Anderson impurity model [873, 874] and its generalizations, namely the many-impurity Anderson models [875]. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots, and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the nonperturbative technique for the thermodynamic Green functions. The method of the irreducible Green functions is used as the basic tool. The subject matter includes the improved interpolating solution of the Anderson model. It will be shown that an interpolating approximation, which simultaneously reproduces the weak-coupling limit up to second order in the interaction strength U and the strong-coupling limit up to second order in the hybridization V (and thus also fulfils the atomic limit), can be formulated self-consistently. This approach offers a new way for the systematic construction of approximate interpolation dynamical solutions of strongly correlated electron systems.

The following sections are included:

• Introduction
• The Spin-Fermion Model
• Quasiparticle Dynamics of the (sp—d)Model
• Spin Dynamics of the sp—d Model: Scattering Regime
• Generalized Mean-Field Green Function
• Uncoupled Subsystems
• Coupled Subsystems
• Effects of Disorder in Diluted Magnetic Semiconductors
• Discussion

The following sections are included:

• Introduction
• Charge and Spin Degrees of Freedom
• Charge Dynamics of the s—d Model: Scattering Regime
• Charge Dynamics of the s—d Model: Bound State Regime
• Magnetic Polaron in GMF
• Damping of the Magnetic Polaron State
• Concluding Remarks

The following sections are included:

• Introduction
• Magnetic Degrees of Freedom
• Microscopic Picture of Magnetism in Materials
  • Heisenberg model
  • Itinerant electron model
  • Hubbard model
  • Multi-band model: model with s–d hybridization
  • Spin–fermion model
• Symmetry and Physics of Magnetism
• Spin Quasiparticle Dynamics
  • Spin dynamics of the Hubbard model
  • Spin dynamics of the SFM
  • Spin dynamics of the multi-band model
• Quasiparticle Excitation Spectra and Neutron Scattering
• Discussion

The following sections are included:

• Introduction
• Gauge Invariance
• Spontaneous Symmetry Breaking
• Goldstone Theorem
• Higgs Phenomenon
• Bogoliubov Quasiaverages in Statistical Mechanics
  • Bogoliubov theorem on the singularity of 1/q²
  • Bogoliubov’s inequality and the Mermin–Wagner theorem
• Broken Symmetries and Condensed Matter Physics
  • Superconductivity
  • Antiferromagnetism
  • Bose systems
• Conclusions and Discussions
• Biography of N. N. Bogoliubov

The following sections are included:

• Introduction
• Emergence Concept
• Emergent Phenomena
• Quantum Mechanics and its Emergent Macrophysics
• Emergent Phenomena in Quantum Condensed Matter Physics
• Discussion

The following sections are included:

• Introduction
• Electron-Lattice Interaction in Condensed Matter Systems
• Modified Tight-Binding Approximation
  • Electron Green function
  • Phonon Green function
  • Renormalization of the electron and phonon spectra
• Renormalized Spectrum of the Mott–Hubbard Insulator
• The Electron–Phonon Spectral Function
• Electron–Lattice Interaction in Disordered Binary Alloys
  • The model
  • Electron Green functions for alloy
  • Green function for lattice vibrations in alloy
  • The configurational averaging
  • The electronic specific heat
• Discussion
• Biography of Herbert Fröhlich

The following sections are included:

• Introduction
• The Microscopic Theory of Superconductivity
  • The Nambu formalism
  • The Eliashberg equations
• Equations of Superconductivity in Wannier Representations
• Strong-Coupling Equations of Superconductivity
• Equations of Superconductivity in Disordered Binary Alloys
  • The model Hamiltonian
  • Electron Green function and mass operator
  • Renormalized lattice Green function
  • Configurational averaging
  • Simplest method of averaging
  • General averaging scheme
  • The random contact model
  • CPA equations for superconductivity in the contact model
  • Linearized equations and transition temperature
• The Physics of Layered Systems and Superconductivity
  • Layered cuprates
  • Crystal structure of cuprates and mercurocuprates
  • The Lawrence–Doniach model
• Discussion

The following sections are included:

• Introduction
• Generalized SFM
• Spin Dynamics of the d–f Model
  • Generalized Mean-field Green Function
  • Dyson Equation for d–f Model
• Self-Energy and Damping
• Charge Dynamics of the d–f Model
• Conclusions

The following sections are included:

• Introduction
• The Electronic Properties of Correlated Systems
• The Model Hamiltonian
• The Effective Hamiltonians
• Coexistence of Spin and Carrier Subsystems
• Competition of Interactions in Kondo Systems
• Dynamics of Carriers in the Spin–Fermion Model
  • Hole dynamics in cuprates
  • Hubbard model and t–J model
  • Hole spectrum of t–J model
  • The Kondo–Heisenberg model
  • Hole dynamics in the Kondo–Heisenberg model
  • Dynamics of spin subsystem
  • Damping of hole quasiparticles
• Concluding Remarks

The following sections are included:

• Introduction
• The Helmholtz Free Energy
• Approximate Calculations of the Helmholtz Free Energy
• The Mean Field Concept
• The Mathematical Tools
• Variational Principle of Bogoliubov
• Applications of the Bogoliubov Variational Principle
• The Variational Schemes and Bounds on Free Energy
• The Hartree–Fock–Bogoliubov Mean Fields
• Method of an Approximating Hamiltonian
• Conclusions

The following sections are included:

• Introduction
• Ensemble Method in the Theory of Irreversibility
• Statistical Mechanics of Irreversibility
• Boltzmann Equation
• The Method of Time Correlation Functions
• Kubo Linear Response Theory
• Fluctuation Theorem and Green–Kubo Relations
• Conditions for Local Equilibrium
• Modified Projection Methods

The following sections are included:

• Nonequilibrium Ensembles
• The NSO Method
• The Relevant Operators
• Construction of the NSO

The following sections are included:

• Hydrodynamic Equations
• The Transport and Kinetic Equations
• System in Thermal Bath: Generalized Kinetic Equations
• System in Thermal Bath: Rate and Master Equations
• A Dynamical Systemin a Thermal Bath
• Schrödinger-type Equation with Damping for a Dynamical System in a Thermal Bath

The following sections are included:

• Damping Effects in a System Interacting with a Thermal Bath
• The Natural Width of Spectral Line of the Atomic System
• Evolution of a System in an Alternating External Field
• Statistical Theory of Spin Relaxation and Diffusion in Solids
  • Dynamics of nuclear spin system
  • Nuclear spin–lattice relaxation
  • Spin diffusion of nuclear magnetic moment
  • Spin diffusion coefficient
  • Stochasticity of spin subsystem
  • Spin diffusion coefficient in dilute alloys
• Other Applications of the NSO Method
• Discussion

In this chapter, the theory of scattering of particles (e.g. neutrons) by statistical medium will be recast for the nonequilibrium statistical medium [1017]. The correlation scattering function of the relevant variables gives rise to a very compact and entirely general expression for the scattering cross-section of interest. The formula obtained by Van Hove [1007, 1010] provides a convenient method of analyzing the properties of slow neutron and light scattering by systems of particles such as gas, liquid or solid in the equilibrium state. Here, the theory of scattering of particles by many-body system will be reformulated and generalized for the case of nonequilibrium statistical medium. A new method of quantum-statistical derivation of the space and time Fourier transforms of the Van Hove correlation function will be formulated. Thus, in place of the usual Van Hove scattering function, a generalized one will be deduced and the result will be shown to be of greater potential utility than those previously given in the literature. This expression gives a natural extension of the familiar Van Hove formula for scattering of slow neutrons for the case in which the system under consideration is in a nonequilibrium state. The feasibility of light- and neutron-scattering experiments to investigate the appropriate problems in real physical systems will be discussed briefly.

In this chapter, some selected approaches to the description of transport properties, mainly electroconductivity, in crystalline and disordered metallic systems will be analyzed. A detailed qualitative theoretical formulation of the electron transport processes in metallic systems within a model approach is given. Generalized kinetic equations which were derived by the method of the nonequilibrium statistical operator (NSO) in Chapter 32 are used. Tight-binding picture and modified tight-binding approximation (MTBA) will be used for describing the electron subsystem and the electron–lattice interaction correspondingly. The low- and high-temperature behavior of the resistivity are discussed in detail. The main objects of discussion are nonmagnetic (or paramagnetic) transition metals and their disordered alloys. The choice of topics and the emphasis on concepts and model approach make it a good method for a better understanding of the electrical conductivity of the transition metals and their disordered binary substitutional alloys, but the formalism developed can be applied (with suitable modification), in principle, to other systems. The approach we used and the results obtained complement the existent theories of the electrical conductivity in metallic systems. We will see that the present study extends the standard theoretical format and calculation procedures in the theories of electron transport in solids.

The following sections are included:

• Bibliography
• Index