Statistical Mechanics of Lattice Systems: Volume 2: Exact, Series and Renormalization Group Methods (Theoretical and Mathematical Physics)

Cover......Page 1
Preface......Page 6
Table of Contents......Page 8
Introduction......Page 14
Thermodynamic Formulae and Variables......Page 15
Statistical Mechanical Formulae......Page 16
The Field Density and Coupling Density Representations......Page 17
Correlation Functions and Symmetry Properties......Page 23
Examples......Page 27
Introduction......Page 28
The Geometry of Phase Transitions......Page 30
Universality Fluctuations and Scaling......Page 34
General Scaling Formulation......Page 40
Logarithmic Singularities......Page 46
Correlation Functions......Page 50
Densities and Response Functions......Page 54
Critical Point and Coexistence Curve......Page 55
Scaling for a Critical Point......Page 59
Mean Field Theory for the Ising Ferromagnet......Page 68
Correlation Scaling at a Critical Point......Page 70
Tricritical Point......Page 71
Scaling for a Tricritical Point......Page 75
Corrections to Scaling......Page 82
Scaling and Universality......Page 83
Finite Size Scaling......Page 88
Conformal Invariance......Page 94
Examples......Page 99
The Ferromagnetic Ising Model......Page 102
Landau Theory for a Critical Point......Page 104
Landau Ginzberg Theory for a Critical Point......Page 106
The Equilibrium Dilute Ising Model......Page 116
The State Potts Model......Page 118
Landau Theory for a Tricritical Point......Page 121
Examples......Page 124
Introduction......Page 126
The Thermodynamic Limit......Page 129
Lower Bounds for Phase Transitions the Peierls Method......Page 130
Lower Bounds for the Simple Lattice Fluid......Page 136
Grand Partition Function Zeros and Phase Transitions......Page 138
Ruelle s Theorem......Page 140
The Yang Lee Circle Theorem......Page 143
Systems with Pair Interactions......Page 145
Transfer Matrices......Page 149
The Wood Method......Page 161
Examples......Page 176
Introduction......Page 180
Spin Representations......Page 182
Parameter Space and Ground States......Page 185
Some Transformations......Page 187
The Weak Graph Transformation......Page 189
Transition Surfaces......Page 190
The Transfer Matrix......Page 195
The Free Energy and Magnetization......Page 199
Critical Behaviour......Page 200
The Spin Representation and the Ising Model Limit......Page 203
The Six Vertex Model as a Special Case......Page 208
The Eight Vertex Model and Universality......Page 211
Examples......Page 213
Introduction......Page 216
The Basic Elements of the Renormalization Group......Page 217
Renormalization Transformations and Weight Functions......Page 219
Fixed Points and the Linear Renormalization Group......Page 224
Free Energy and Densities......Page 227
Decimation of the One Dimensional Ising Model......Page 228
Decimation in Two Dimensions......Page 235
Lower Bound and Upper Bound Approximations......Page 238
The Cumulant Approximation......Page 242
Bond Moving Approximations......Page 245
Finite Lattice Approximations......Page 248
Variational Approximations......Page 252
Phenomenological Renormalization......Page 254
Other Renormalization Group Methods......Page 260
Examples......Page 261
Introduction......Page 264
The Analysis of Series......Page 267
Ising Model......Page 271
Ising Model......Page 275
The Linked Cluster Expansion......Page 283
Applications of the Linked Cluster Expansion......Page 290
Finite Methods......Page 303
Results and Analysis......Page 308
Examples......Page 314
Introduction......Page 316
The Dimer Partition Function......Page 317
The Modi ed KDP Model Equivalence......Page 330
The Ising Model Equivalence......Page 333
K Type and O Type Transitions......Page 337
The Chain Conformal Transition......Page 339
Examples......Page 345
A.1 Fourier Transforms in d Dimensions......Page 348
A.2 The Conformal Group......Page 353
A.3 Group Representation Theory......Page 355
A.4
 Some Transformations in the Complex Plane......Page 373
A.5 Algebraic Functions......Page 376
A.6 Elliptic Integrals Functions and Nome Series......Page 379
A.7
 Lattices and Graphs......Page 383
A.8 The Weak Graph Transformation......Page 394
A.9
 The Generalized Moment Cumulant Relations......Page 399
A.10 Kastelyn's Theorem......Page 402
A.11 Determinants of Cyclic Matrices......Page 407
A.12 The T Matrix......Page 409
References and Author Index......Page 414
Subject Index......Page 436