دانلود کتاب Statistical Field Theory - An Introduction to Exactly Solved Models in Statistical Physics

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Cover Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics Copyright Preface to the first edition Preface to the second edition Structure of the book Acknowledgements Contents Part 1: Preliminary Notions Chapter 1: Introduction 1.1 Phase Transitions 1.1.1 Competitive Principles 1.1.2 Partition Function 1.1.3 Order Parameters 1.1.4 Correlation Functions 1.1.5 Critical Exponents 1.1.6 Scaling Laws 1.1.7 Dimensionality of Space and Order Parameters 1.2 The Ising Model 1.3 Ernst Ising Appendix 1.A. Ensembles in Classical Statistical Mechanics Appendix 1.B. Ensembles in Quantum Statistical Mechanics References Problems Chapter 2: One-dimensional Systems 2.1 Recursive Approach 2.2 Transfer Matrix 2.2.1 Periodic Boundary Conditions 2.2.2 Other Boundary Conditions: Boundary States 2.3 Series Expansions 2.4 Critical Exponents and Scaling Laws 2.5 The Potts Model 2.6 Models with O(n) Symmetry 2.7 Models with Zn Symmetry 2.8 Feynman Gas Appendix 2.A. Special Functions The (z) function The Bessel functions Iν(x) The Bessel functions Kν(x) Appendix 2.B. n-dimensional Solid Angle Appendix 2.C. The Four-colour Problem References Problems Chapter 3: Approximate Solutions 3.1 Mean Field Theory of the Ising Model 3.2 Mean Field Theory of the Potts Model 3.3 Bethe–Peierls Approximation 3.4 The Gaussian Model 3.5 The spherical model Appendix 3.A. The Saddle Point Method Appendix 3.B. Brownian Motion on a Lattice References Problems Part 2: Bi-dimensional Lattice Models Chapter 4: Duality of the Two-dimensional Ising Model 4.1 Peierls Argument 4.2 Duality Relation in Square Lattices 4.2.1 High Temperature Series Expansion 4.2.2 Low-temperature Series Expansion 4.2.3 Self-duality 4.3 Duality Relation: Hexagonal and Triangular Lattices 4.4 Star-triangle Identity 4.5 Ising Model Critical Temperature: Triangle and Hexagonal Lattices 4.6 Duality in Two Dimensions 4.6.1 Self-duality of the p-state Model 4.6.2 Duality Relation between XY Model and SOS Model Appendix 4.A. Numerical Series Appendix 4.B. Poisson Sum Formula References Problems Chapter 5: Combinatorial Solutions of the Ising Model 5.1 Combinatorial Approach 5.1.1 Partition Function 5.1.2 Correlation Function and Magnetization 5.2 Dimer Method 5.2.1 Dimers on a Square Lattice 5.2.2 Dimer Formulation of the Ising Model References Problems Chapter 6: Transfer Matrix of the Two-dimensional Ising Model 6.1 Baxter’s Approach 6.1.1 Commutativity of the Transfer Matrices 6.1.2 Commutativity of the Transfer Matrices: Graphical Proof 6.1.3 Functional Equations and Symmetries 6.1.4 Functional Equations for the Eigenvalues 6.2 Eigenvalue Spectrum at the Critical Point 6.3 Away from the Critical Point 6.4 Yang–Baxter Equation and R-matrix 6.4.1 Six-vertex model References Problems Part 3: Quantum Field Theory and Conformal Invariance Chapter 7: Quantum Field Theory 7.1 Motivations 7.2 Order Parameters and Lagrangian 7.3 Field Theory of the Ising Model 7.4 Correlation Functions and Propagator 7.5 Perturbation Theory and Feynman diagrams 7.6 Legendre Transformation and Vertex Functions 7.7 Spontaneous Symmetry Breaking and Multi-criticality 7.7.1 Universality Class of the Ising model 7.7.2 Universality Class of the Tricritical Ising Model 7.7.3 Multicritical Points 7.8 Renormalization 7.9 Field Theory in Minkowski Space 7.10 Particles 7.11 Correlation Functions and Scattering Processes Appendix 7.A. Feynman Path Integral Formulation Appendix 7.B. Relativistic Invariance Appendix 7.C. Noether Theorem References Problems Chaoter 8: Renormalization Group 8.1 Introduction 8.2 Reducing the Degrees of Freedom 8.3 Transformation Laws and Effective Hamiltonians 8.4 Fixed Points 8.5 The Ising Model 8.6 The Gaussian Model 8.7 Operators and Quantum Field Theory 8.8 Functional Form of the Free Energy 8.9 Critical Exponents and Universal Ratios 8.10 β-Functions References Problems Chapter 9: Fermionic Formulation of the Ising Model 9.1 Introduction 9.2 Transfer Matrix and Hamiltonian Limit 9.3 Order and Disorder Operators 9.4 Perturbation Theory 9.5 Expectation Values of Order and Disorder Operators 9.6 Diagonalization of the Hamiltonian 9.7 Dirac Equation References Problems Chapter 10: Conformal Field Theory 10.1 Introduction 10.2 The Algebra of Local Fields 10.3 Conformal Invariance 10.3.1 Conformal Transformations in D Dimensions 10.3.2 Polyakov’s Theorem 10.4 Quasi-primary Fields 10.5 Two-dimensional Conformal Transformations 10.6 Ward Identity and Primary Fields 10.7 Central Charge and Virasoro Algebra 10.7.1 Example 1. Free Neutral Fermion 10.7.2 Example 2. Free Bosonic Field 10.8 Representation Theory 10.8.1 Representation Theory: the Space of the Conformal States 10.8.2 Representation Theory: The Space of Conformal Fields 10.9 Hamiltonian on a Cylinder Geometry and Casimir Effect 10.10 Entanglement Entropy Appendix 10.A. Moebius Transformations References Problems Chapter 11: Minimal Conformal Models 11.1 Introduction 11.2 Null Vectors and Kac Determinant 11.3 Unitary Representations 11.4 Minimal Models 11.4.1 Kac Table 11.4.2 Differential Equations 11.4.3 Operator Product Expansion and Fusion Rules 11.4.4 Verlinde Algebra 11.5 Coulomb Gas 11.5.1 Free Theory of a Bosonic Field 11.5.2 Modified Coulomb Gas 11.5.3 Screening Operators 11.5.4 Correlation Functions 11.6 Landau–Ginzburg Formulation 11.7 Modular Invariance 11.7.1 Torus Geometry 11.7.2 Partition Function and Characters Appendix 11.A. Hypergeometric functions References Problems Chapter 12: Conformal Field Theory of Free Bosonic and Fermionic Fields 12.1 Introduction 12.2 Conformal Field Theory of Free Bosonic Fields 12.2.1 Quantization of the Bosonic Field 12.2.2 Vertex Operators 12.2.3 Free Bosonic Field on a Torus 12.3 Conformal Field Theory of a Free Fermionic Field 12.3.1 Quantization of the Free Majorana Fermion 12.3.2 Fermions on a Torus 12.4 Bosonization 12.4.1 Bosonization Rules References Problems Chapter 13: Conformal Field Theories with Extended Symmetries 13.1 Introduction 13.2 Superconformal Models 13.3 Parafermion Models 13.3.1 Relation to Lattice Models 13.4 Kac–Moody Algebra 13.4.1 Virasoro Operators and Sugawara Formula 13.4.2 Maximal Weights 13.4.3 Wess–Zumino–Witten Models 13.5 Conformal Models as Cosets 13.5.1 Relation with parafermions Appendix 13.A. Lie Algebra References Problems Chapter 14: The Arena of Conformal Models 14.1 Introduction 14.2 The Ising Model 14.2.1 Operator Product Expansion and Correlation Functions 14.2.2 Coset Constructions and E8 Algebra 14.2.3 Characters and Partition Function 14.3 The Universality Class of the Tricritical Ising Model 14.4 3-state Potts Model 14.5 The Yang–Lee Model 14.6 Conformal Models with O(n) Symmetry References Problems Part 4: Away from Criticality Chapter 15: In the Vicinity of the Critical Points 15.1 Introduction 15.2 Conformal Perturbation Theory 15.3 Example: The Two-point Function of the Yang–Lee model 15.4 Renormalization Group and β-functions 15.5 c-theorem 15.6 Applications of the c theorem 15.6.1 Minimal Conformal Models Mp perturbed by the 1,3 Operator 15.6.2 Ising model at temperature T = Tc 15.6.3 A Lagrangian theory: the Sine–Gordon model 15.7  theorem References Chapter 16: Integrable Quantum Field Theories 16.1 Introduction 16.2 The Sinh–Gordon Model 16.3 The Sine–Gordon Model 16.4 The Bullogh–Dodd Model 16.5 Integrability versus Non-integrability 16.6 The Toda Field Theories 16.6.1 A(1)n Series 16.6.2 D(1)n Series 16.6.3 En Series 16.7 Toda Field Theories with Imaginary Coupling Constant 16.8 Deformation of Conformal Conservation Laws 16.8.1 Operator Product Expansion 16.8.2 Integrals of Motion of the Identity Family 16.8.3 Counting Method 16.8.4 Examples 16.9 Multiple Deformations of Conformal Field Theories 16.9.1 The Tricritical Ising Model 16.9.2 The Ising Model References Problems Chapter 17: S-matrix Theory 17.1 Analytic Scattering Theory 17.1.1 General Properties 17.1.2 Two-body Scattering Process 17.2 General Properties of Purely Elastic Scattering Matrices 17.2.1 Rapidity variable and asymptotic states 17.2.2 Conserved Charges 17.2.3 Elasticity in the Scattering Processes 17.2.4 Factorization of the Scattering Processes 17.3 Unitarity and Crossing Invariance Equations 17.4 Analytic Structure and Bootstrap Equations 17.5 Conserved Charges and Consistency Equations 17.5.1 Non-degenerate Bootstrap Systems Appendix 17.A. Historical Developments of the S-matrix Theory Appendix 17.B. Scattering Processes in Quantum Mechanics Appendix 17.C. n-particle Phase Space References Problems Chapter 18: Exact S-Matrices 18.1 Yang–Lee and Bullogh–Dodd Models 18.2 1,3 Integrable Deformation of the Conformal Minimal Models M2,2n+3 18.3 Multiple Poles 18.4 S-Matrices of the Ising Model 18.4.1 Thermal deformation of the Ising Model 18.4.2 Magnetic Deformation of the Ising Model 18.5 The Tricritical Ising Model at T = Tc 18.6 Thermal Deformation of the 3-state Potts Model 18.6.1 Thermal Deformation of the 3-state Tricritical Potts Model 18.7 General Expression Toda Field Theories 18.8 Non-relativistic Limit of Toda Field Theories 18.9 Models with Internal O(n) Invariance 18.9.1 n > 2 18.9.2 n < 2 18.10 S-matrix of the Sine-Gordon Model 18.11 S-Matrices for 1,3, 1,2, 2,1 Deformation of Minimal Models 18.11.1 Quantum Group Symmetry of the Sine–Gordon 18.11.2 Restricted Sine–Gordon model 18.11.3 Quantum Group Symmetry of the Bullough–Dodd Model 18.12 Elastic SUSY S-matrix References Problems Chpater 19: Form Factors and Correlation Functions 19.1 General Properties of the Form Factors 19.1.1 Faddeev–Zamolodchikov Algebra 19.1.2 Form Factors 19.2 Watson’s Equations 19.3 Recursive Equations 19.4 The Operator Space 19.5 Correlation Functions 19.6 Form Factors of the Stress-energy Tensor 19.7 Vacuum Expectation Values 19.8 Ultraviolet Limit 19.9 The Ising Model at T =Tc 19.9.1 The Energy Operator 19.9.2 Magnetization Operators 19.9.3 The Painlevé Equation 19.10 Form Factors of the Sinh–Gordon Model 19.10.1 Minimal Form Factor 19.10.2 Recursive Equations 19.10.3 General Properties of the Qn Solutions 19.10.4 The Elementary Solutions 19.11 The Ising Model in a Magnetic Field References Problems Part 5: Finite Size Effects Chapter 20: Thermodynamic Bethe Ansatz 20.1 Introduction 20.2 Casimir Energy 20.3 Bethe Relativistic Wave Function 20.3.1 Selection Rules 20.4 Derivation of Thermodynamics 20.5 The Meaning of Pseudo-energy 20.6 Infrared and Ultraviolet Limits 20.7 The Coefficient of Bulk Energy 20.8 The General Form of the TBA Equations 20.9 The Exact Relation λ(m) 20.10 Examples 20.10.1 Yang–Lee 20.10.2 The Ising Model in a Magnetic Field 20.10.3 The Tricritical Ising Model 20.11 Thermodynamics of the Free Field Theories 20.12 L-channel Quantization 20.13 LeClair–Mussardo formula References Problems Chapter 21: Boundary Field Theory 21.1 Introduction 21.2 Stress-energy Tensor in Boundary CFT 21.3 Conformal Boundary Operators 21.4 Conformal Boundary States 21.4.1 Boundary Entropy 21.5 Operator Product Expansion Involving a Boundary Operator 21.6 Massive Integrable Boundary Field Theory 21.7 Boundary States 21.8 Massive Boundary Ising Model 21.9 Correlation Functions References Problems Part 6: Non-Integrable Aspects Chapter 22: Form Factor Perturbation Theor 22.1 Breaking Integrability 22.2 Multiple Deformations of the Conformal Field Theories 22.3 Form Factor Perturbation Theory 22.4 First-order Perturbation Theory 22.5 Non-locality and Confinement of the Excitations 22.6 Multi-frequency Sine–Gordon Model 22.6.1 The Generalized Ashkin–Teller Model References Problems Chapter 23: Particle Spectrum by Semi-classicalMethods 23.1 Introduction 23.2 Kinks 23.3 A Semi-classical Formula for the Kink Matrix Elements 23.4 Universal Mass Formula 23.5 Symmetric Wells 23.6 Asymmetric Wells 23.7 Double Sine–Gordon Model 23.7.1 Dynamics of Long and Short Kinks 23.7.2 The importance of small kinks References Problems Chapter 24: Interacting Fermions and Supersymmetric Models 24.1 Introduction 24.2 Fermion in a Bosonic Background 24.3 The Fermionic Bound States in T = 0 Sector 24.4 Symmetric Wells 24.5 Supersymmetric Theory 24.6 General Results in SUSY Theories 24.7 Integrable SUSY Models 24.8 Non-integrable Multi-frequency Super Sine-Gordon Models 24.9 Phase Transition and Meta-stable States 24.10 Summary Refernces Problems Chapter 25: Truncated Hilbert Space Approach 25.1 Truncated Hamiltonians of Quantum Mechanics 25.1.1 Harmonic Oscillator 25.1.2 Basis of the Truncated Hamiltonian 25.2 Truncated Hamiltonian of the Deformed Conformal Models 25.2.1 General features of the finite-size energy levels 25.2.2 Effects of truncation 25.3 Finite-size Mass Corrections 25.4 The Scaling Region of the Ising Model 25.4.1 Analysis of the Ising Model through FFPT 25.4.2 Analysis of the Ising Model through THSA References Problems Index