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

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Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics
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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