Contents -- Part I: Preliminary Notions -- 1 Introduction -- 1.1 Phase Transitions -- 1.2 The Ising Model -- 1A: Ensembles in Classical Statistical Mechanics -- 1B: Ensembles in Quantum Statistical Mechanics -- Problems -- 2 One-dimensional Systems -- 2.1 Recursive Approach -- 2.2 Transfer Matrix -- 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 Z[sub(n)] Symmetry -- 2.8 Feynman Gas -- 2A: Special Functions -- 2B: n-dimensional Solid Angle -- 2C: The Four-color Problem -- Problems -- 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 -- 3A: The Saddle Point Method -- 3B: Brownian Motion on a Lattice -- Problems -- Part II: Bidimensional Lattice Models -- 4 Duality of the Two-dimensional Ising Model -- 4.1 Peierls's Argument -- 4.2 Duality Relation in Square Lattices -- 4.3 Duality Relation between Hexagonal and Triangular Lattices -- 4.4 Star-Triangle Identity -- 4.5 Critical Temperature of Ising Model in Triangle and Hexagonal Lattices -- 4.6 Duality in Two Dimensions -- 4A: Numerical Series -- 4B: Poisson Resummation Formula -- Problems -- 5 Combinatorial Solutions of the Ising Model -- 5.1 Combinatorial Approach -- 5.2 Dimer Method -- Problems -- 6 Transfer Matrix of the Two-dimensional Ising Model -- 6.1 Baxter's Approach -- 6.2 Eigenvalue Spectrum at the Critical Point -- 6.3 Away from the Critical Point -- 6.4 Yang-Baxter Equation and R-matrix -- Problems -- Part III: Quantum Field Theory and Conformal Invariance -- 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 Multicriticality -- 7.8 Renormalization -- 7.9 Field Theory in Minkowski Space -- 7.10 Particles -- 7.11 Correlation Functions and Scattering Processes -- 7A: Feynman Path Integral Formulation -- 7B: Relativistic Invariance -- 7C: Noether's Theorem -- Problems -- 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 -- Problems -- 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 -- Problems -- 10 Conformal Field Theory -- 10.1 Introduction -- 10.2 The Algebra of Local Fields -- 10.3 Conformal Invariance -- 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 -- 10.8 Representation Theory -- 10.9 Hamiltonian on a Cylinder Geometry and the Casimir Effect -- 10A: Moebius Transformations -- Problems -- 11 Minimal Conformal Models -- 11.1 Introduction -- 11.2 Null Vectors and Kac Determinant -- 11.3 Unitary Representations -- 11.4 Minimal Models -- 11.5 Coulomb Gas -- 11.6 Landau-Ginzburg Formulation -- 11.7 Modular Invariance -- 11A: Hypergeometric Functions -- Problems -- 12 Conformal Field Theory of Free Bosonic and Fermionic Fields -- 12.1 Introduction.

12.2 Conformal Field Theory of a Free Bosonic Field -- 12.3 Conformal Field Theory of a Free Fermionic Field -- 12.4 Bosonization -- Problems -- 13 Conformal Field Theories with Extended Symmetries -- 13.1 Introduction -- 13.2 Superconformal Models -- 13.3 Parafermion Models -- 13.4 Kac-Moody Algebra -- 13.5 Conformal Models as Cosets -- 13A: Lie Algebra -- Problems -- 14 The Arena of Conformal Models -- 14.1 Introduction -- 14.2 The Ising Model -- 14.3 The Universality Class of the Tricritical Ising Model -- 14.4 Three-state Potts Model -- 14.5 The Yang-Lee Model -- 14.6 Conformal Models with O(n) Symmetry -- Problems -- Part IV: Away from Criticality -- 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 -- 15.7 Δ-theorem -- 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 -- 16.7 Toda Field Theories with Imaginary Coupling Constant -- 16.8 Deformation of Conformal Conservation Laws -- 16.9 Multiple Deformations of Conformal Field Theories -- Problems -- 17 S-Matrix Theory -- 17.1 Analytic Scattering Theory -- 17.2 General Properties of Purely Elastic Scattering Matrices -- 17.3 Unitarity and Crossing Invariance Equations -- 17.4 Analytic Structure and Bootstrap Equations -- 17.5 Conserved Charges and Consistency Equations -- 17A: Historical Development of S-Matrix Theory -- 17B: Scattering Processes in Quantum Mechanics -- 17C: n-particle Phase Space -- Problems -- 18 Exact S-Matrices -- 18.1 Yang-Lee and Bullogh-Dodd Models.

18.2 Φ[sub(1,3)] Integrable Deformation of the Conformal Minimal Models M[sub(2,2n+3)] -- 18.3 Multiple Poles -- 18.4 S-Matrices of the Ising Model -- 18.5 The Tricritical Ising Model at T≠Tc -- 18.6 Thermal Deformation of the Three-state Potts Model -- 18.7 Models with Internal O(n) Invariance -- 18.8 S-Matrix of the Sine-Gordon Model -- 18.9 S-Matrices for Φ[sub(1,3)], Φ[sub(1,2)], Φ[sub(2,1) Deformation of Minimal Models -- Problems -- 19 Thermodynamical Bethe Ansatz -- 19.1 Introduction -- 19.2 Casimir Energy -- 19.3 Bethe Relativistic Wave Function -- 19.4 Derivation of Thermodynamics -- 19.5 The Meaning of the Pseudo-energy -- 19.6 Infrared and Ultraviolet Limits -- 19.7 The Coefficient of the Bulk Energy -- 19.8 The General Form of the TBA Equations -- 19.9 The Exact Relation λ(m) -- 19.10 Examples -- 19.11 Thermodynamics of the Free Field Theories -- 19.12 L-channel Quantization -- Problems -- 20 Form Factors and Correlation Functions -- 20.1 General Properties of the Form Factors -- 20.2 Watson's Equations -- 20.3 Recursive Equations -- 20.4 The Operator Space -- 20.5 Correlation Functions -- 20.6 Form Factors of the Stress-Energy Tensor -- 20.7 Vacuum Expectation Values -- 20.8 Ultraviolet Limit -- 20.9 The Ising Model at T≠Tc -- 20.10Form Factors of the Sinh-Gordon Model -- 20.11The Ising Model in a Magnetic Field -- Problems -- 21 Non-Integrable Aspects -- 21.1 Multiple Deformations of the Conformal Field Theories -- 21.2 Form Factor Perturbation Theory -- 21.3 First-order Perturbation Theory -- 21.4 Non-locality and Confinement -- 21.5 The Scaling Region of the Ising Model -- Problems -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Y -- Z.