Preface v 1 Basic Statistical Mechanics 1.1 Phase transitions and critical points 1 1.2 The scaling hypothesis 4 1.3 Universality 7 1.4 The partition function 8 1.5 Approximation methods 9 1.6 Exactly solved models 11 1.7 The general Ising model 14 1.8 Nearest-neighbour Ising model 21 1.9 The lattice gas 24 1.10 The van der Waals fluid and classical exponents 30 2 The One-dimensional Ising Model 2.1 Free energy and magnetization 32 2.2 Correlations 35 2.3 Critical behaviour near T = 0 37 3 The Mean Field Model 3.1 Thermodynamic properties 39 3.2 Phase transition 42 3.3 Zero-field properties and critical exponents 44 3.4 Critical equation of state 45 3.5 Mean field lattice gas 46 4 Ising Model on the Bethe Lattice 4.1 The Bethe lattice 47 4.2 Dimensionality 49 4.3 Recurrence relations for the central magnetization 49 4.4 The limit n--> infinity 51 4.5 Magnetization as a function of H 53 4.6 Free energy 55 4.7 Low-temperature zero-field results 56 4.8 Critical behaviour 57 4.9 Anisotropic model 58 5 The Spherical Model 5.1 Formulation of the model 60 5.2 Free energy 61 5.3 Equation of state and internal energy 64 5.4 The function g'(z) 65 5.5 Existence of a critical point for d > 2 66 5.6 Zero-field properties: exponents ?, ?, ?, ?' 68 5.7 Critical equation of state 70 6 Duality and Star Triangle Transformations of Planar Ising Models 6.1 General comments on two-dimensional models 72 6.2 Duality relation for the square lattice Ising model 73 6.3 Honeycomb-triangular duality 78 6.4 Star-triangle relation 80 6.5 Triangular triangular duality 86 7 Square-Lattice Ising Model 7.1 Historical introduction 88 7.2 The transfer matrices V, W 89 7.3 Two significant properties of V and W 91 7.4 Symmetry relations 95 7.5 Commutation relations for transfer matrices 96 7.6 Functional relation for the eigenvalues 97 7.7 Eigenvalues ? for T = Tc 98 7.8 Eigenvalues ? for T less than Tc 101 7.9 General expressions for the eigenvalues 108 7.10 Next-largest eigenvalues: interfacial tension, correlation length and magnetization for T less than Tc 111 7.11 Next-largest eigenvalue and correlation length for T>Tc 119 7.12 Critical behaviour 120 7.13 Parametrized star-triangle relation 122 7.14 The dimer problem 124 8 Ice-Type Models 8.1 Introduction 127 8.2 The transfer matrix 130 8.3 Line-conservation 131 8.4 Eigenvalues for arbitrary n 138 8.5 Maximum Eigenvalue: location of z1,...,zn 140 8.6 The case ? > 1 143 8.7 Thermodynamic limit for ? less than 1 143 8.8 Free energy for - 1 less than ? less than 1 145 8.9 Free energy for ? less than - 1 148 8.10 Classification of phases 150 8.11 Critical singularities 156 8.12 Ferroelectric model in a field 160 8.13 Three-colourings of the square lattice 165 9 Alternative Way of Solving the Ice-Type Models 9.1 Introduction. 180 9.2 Commuting transfer matrices 180 9.3 Equations for the eigenvalues 181 9.4 Matrix function relation that defines the eigenvalues 182 9.5 Summary of the relevant matrix properties 184 9.6 Direct derivation of the matrix properties: commutation 185 9.7 Parametrization in terms of entire functions 190 9.8 The matrix Q(v) 192 9.9 Values of ?, ?, upsilon 200 10 Square Lattice Eight-Vertex Model 10.1 Introduction 202 10.2 Symmetries 204 10.3 Formulation as an Ising model with two- and four-spin interactions 207 10.4 Star triangle relation 210 10.5 The matrix Q(upsilon) 215 10.6 Equations for the eigenvalues of V(upsilon) 222 10.7 Maximum eigenvalue: location of upsilon1,...,upsilonn 224 10.8 Calculation of the free energy 228 10.9 The Ising case 237 10.10 Other thermodynamic properties 239 10.11 Classification of phases 245 10.12 Critical singularities 248 10.13 An equivalent Ising model 255 10.14 The XYZ chain 258 10.15 Summary of definitions of ?, ?, k, ?, upsilon, q, x, z, p, ?, w 267 10.16 Special cases 269 10.17 An exactly solvable inhomogeneous eight-vertex model 272 11 Kagome Lattice Eight-Vertex Model 11.1 Definition of the model 276 11.2 Conversion to a square-lattice model 281 11.3 Correlation length and spontaneous polarization 284 11.4 Free energy 285 11.5 Formulation as a triangular-honeycomb Ising model with two- and four-spin interactions 286 11.6 Phases 293 11.7 K" = 0: The triangular and honeycomb Ising models 294 11.8 Explicit expansions of the Ising model results 300 11.9 Thirty-two vertex model 309 11.10 Triangular three-spin model 314 12 Potts and Ashkin Teller Models 12.1 Introduction and definition of the Potts model 322 12.2 Potts model and the dichromatic polynomial 323 12.3 Planar graphs: equivalent ice-type model 325 12.4 Square-lattice Potts model 332 12.5 Critical square-lattice Potts model 339 12.6 Triangular-lattice Potts model 345 12.7 Combined formulae for all three planar lattice Potts models 350 12.8 Critical exponents of the two-dimensional Potts model 351 12.9 Square-lattice Ashkin Teller model 353 13 Corner Transfer Matrices 13.1 Definitions 363 13.2 Expressions as products of operators 369 13.3 Star triangle relation 370 13.4 The infinite lattice limit 376 13.5 Eigenvalues of the CTMs 377 13.6 Inversion properties: relation for K(u) 382 13.7 Eight-vertex model 385 13.8 Equations for the CTMs 389 14 Hard Hexagon and Related Models 14.1 Historical background and principal results 402 14.2 Hard square model with diagonal interactions 409 14.3 Free energy 420 14.4 Sub-lattice densities and the order parameter R 426 14.5 Explicit formulae for the various cases: the Rogers Ramanujan identities 432 14.6 Alternative expressions for the k, p, R 443 14.7 The hard hexagon model 448 14.8 Comments and speculations 452 14.9 Acknowledgements 454 15 Elliptic Functions 15.1 Definitions 455 15.2 Analyticity and periodicity 456 15.3 General theorems 458 15.4 Algebraic identities 460 15.5 Differential and integral identities 464 15.6 Landen transformation 466 15.7 Conjugate modulus 467 15.8 Poisson summation formula 468 15.9 Series expansions of the theta functions 469 15.10 Parametrization of symmetric biquadratic relations 471 16 Subsequent Developments 16.1 Introduction 474 16.2 Three-dimensional models 474 16.3 Chiral Potts model 475 References 485 Supplementary References 493 Index 495