ISBN: 3-540-64436-9
TITLE: Statistical Mechanics of Lattice Systems
AUTHOR: Lavis, David A.; Bell, George M.
TOC:

1. Thermodynamics and Statistical Mechanics 1 
1.1 Introduction 1 
1.2 Thermodynamic Formulae and Variables 2 
1.3 Statistical Mechanical Formulae 3 
1.4 The Field-Density and Coupling-Density Representations 4 
1.4.1 Thermodynamic Formalism 5 
1.4.2 Lattice Systems 8 
1.5 Correlation Functions and Symmetry Properties 10 
1.5.1 Correlation Functions 10 
1.5.2 Symmetry Properties 12 
Examples 14 
2. Phase Transitions and Scaling Theory 15 
2.1 Introduction 15 
2.2 The Geometry of Phase Transitions 17 
2.2.1 A Two-Dimensional Phase Space 18 
2.2.2 A Three-Dimensional Phase Space 20 
2.3 Universality, Fluctuations and Scaling 21 
2.3.1 Universality 22 
2.3.2 Kadanoff's Scaling Method for the Ising Model 24 
2.4 General Scaling Formulation 27 
2.4.1 The Kadano, Scaling Hypothesis 27 
2.4.2 Approaches to the Transition Region 30 
2.4.3 First-Order Transitions 31 
2.4.4 Effective Exponents 32 
2.5 Logarithmic Singularities 33 
2.5.1 The Nightingale'T Hooft Scaling Hypothesis 33 
2.5.2 Constraints on Scaling 34 
2.5.3 Approaches to the Transition Region 35 
2.6 Correlation Functions 37 
2.6.1 Scaling Operators and Dimensions 37 
2.6.2 Variable Scaling Exponents 40 
2.7 Densities and Response Functions 41 
2.8 Critical Point and Coexistence Curve 42 
2.8.1 Response Functions 44 
2.8.2 Critical Exponents 45 
2.8.3 Exponent Inequalities 46 
2.9 Scaling for a Critical Point 46 
2.9.1 Scaling Fields for the Critical Point 47 
2.9.2 Approaches to the Critical Point 48 
2.9.3 Experimental Variables 50 
2.9.4 The Density and Response Functions 51 
2.9.5 Asymptotic Forms 51 
2.9.6 Critical Exponents and Scaling Laws 52 
2.9.7 Scaling for the Coexistence Curve 54 
2.10 Mean-Field Theory for the Ising Ferromagnet 55 
2.11 Correlation Scaling at a Critical Point 57 
2.12 Tricritical Point 58 
2.13 Scaling for a Tricritical Point 62 
2.13.1 Scaling Fields for the Tricritical Point 62 
2.13.2 Tricritical Exponents and Scaling Laws 64 
2.13.3 Connected Transition Regions 66 
2.14 Corrections to Scaling 69 
2.15 Scaling and Universality 70 
2.16 Finite-Size Scaling 75 
2.16.1 The Finite-Size Scaling Field 76 
2.16.2 The Shift and Rounding Exponents 78 
2.16.3 Universality and Finite-Size Scaling 80 
2.17 Conformal Invariance 81 
2.17.1 From Scaling to the Conformal Group 82 
2.17.2 Correlation Functions for d >= 2 83 
2.17.3 Universal Amplitudes for d = 2 84 
Examples 86 
3. Landau and LandauGinzburg Theory 89 
3.1 The Ferromagnetic Ising Model 89 
3.2 Landau Theory for a Critical Point 91 
3.3 LandauGinzberg Theory for a Critical Point 93 
3.3.1 The Gaussian Approximation 94 
3.3.2 Gaussian Critical Exponents 97 
3.4 The Equilibrium Dilute Ising Model 103 
3.5 The 3-State Potts Model 105 
3.6 Landau Theory for a Tricritical Point 108 
Examples 111 
4. Algebraic Methods in Statistical Mechanics 113 
4.1 Introduction 113 
4.2 The Thermodynamic Limit 116 
4.3 Lower Bounds for Phase Transitions: the Peierls Method 117 
4.4 Lower Bounds for the Simple Lattice Fluid 123 
4.5 Grand Partition Function Zeros and Phase Transitions 125 
4.6 Ruelle's Theorem 127 
4.7 The YangLee Circle Theorem 130 
4.8 Systems with Pair Interactions 132 
4.9 Transfer Matrices 136 
4.9.1 The Partition Function 137 
4.9.2 Boundary Conditions 140 
4.9.3 The Limit N_2 --> Infinity 140 
4.9.4 Correlation Functions 141 
4.9.5 Correlation Lengths and Phase Transitions 145 
4.10 The Wood Method 148 
4.10.1 Evolution of Partition Function Zeros 149 
4.10.2 Connection Curves and Cross-Block Curves 151 
4.10.3 The Spin- Square Lattice Ising Model 154 
4.10.4 Critical Points and Exponents 159 
Examples 163 
5. The Eight-Vertex Model 167 
5.1 Introduction 167 
5.2 Spin Representations 169 
5.3 Parameter Space and Ground States 172 
5.4 Some Transformations 174 
5.5 The Weak-Graph Transformation 176 
5.6 Transition Surfaces 177 
5.7 The Transfer Matrix 182 
5.8 The Free Energy and Magnetization 186 
5.9 Critical Behaviour 187 
5.10 The Spin Representation and the Ising Model Limit 190 
5.10.1 The Isotropic Ising Model With a Four-Spin Coupling 191 
5.10.2 The Jngling Spin Representation 192 
5.10.3 The Isotropic Ising Model 
Without a Four-Spin Coupling 192 
5.11 The Six-Vertex Model as a Special Case 195 
5.11.1 Low-Temperature Regions (i) and (ii) 195 
5.11.2 Low-Temperature Region (iii) 196 
5.11.3 High-Temperature Regions (i), (ii) and (iii) 197 
5.12 The Eight-Vertex Model and Universality 198 
Examples 200 
6. Real-Space Renormalization Group Theory 203 
6.1 Introduction 203 
6.2 The Basic Elements of the Renormalization Group 204 
6.3 Renormalization Transformations and Weight Functions 206 
6.4 Fixed Points and the Linear Renormalization Group 211 
6.5 Free Energy and Densities 214 
6.6 Decimation of the One-Dimensional Ising Model 215 
6.7 Decimation in Two Dimensions 222 
6.8 Lower-Bound and Upper-Bound Approximations 225 
6.8.1 An Upper-Bound Method 226 
6.8.2 A Lower-Bound Method 228 
6.9 The Cumulant Approximation 229 
6.10 Bond Moving Approximations 232 
6.11 Finite-Lattice Approximations 235 
6.12 Variational Approximations 239 
6.13 Phenomenological Renormalization 241 
6.13.1 The Square Lattice Ising Model 244 
6.13.2 Other Models 245 
6.13.3 More Than One Coupling 246 
6.14 Other Renormalization Group Methods 247 
Examples 248 
7. Series Methods 251 
7.1 Introduction 251 
7.2 The Analysis of Series 254 
7.2.1 The Ratio Method 254 
7.2.2 Pad Approximants 256 
7.2.3 The Differential Approximant Method 258 
7.3 Low-Temperature Series for the Spin- Ising Model 258 
7.4 High-Temperature Series for the Spin- Ising Model 262 
7.4.1 The Free Energy 262 
7.4.2 Susceptibility Series 264 
7.4.3 Coefficient Relations for Susceptibility Series 266 
7.5 The Linked-Cluster Expansion 270 
7.5.1 Multi-Bonded Graphs 270 
7.5.2 Connected Graphs and Stars 273 
7.5.3 Moments, Cumulants and Finite Clusters 275 
7.6 Applications of the Linked-Cluster Expansion 277 
7.6.1 The General-s Ising Model 277 
7.6.2 D-Vector Models 279 
7.6.3 The Classical Heisenberg Model 280 
7.6.4 The Quantum Heisenberg Model 283 
7.6.5 Correlations and Susceptibility 286 
7.7 Finite Methods 290 
7.7.1 The Finite-Cluster Method 291 
7.7.2 The Finite-Lattice Method 293 
7.8 Results and Analysis 295 
7.8.1 High-Temperature Series for Spin- Ising 
and Potts Models 296 
7.8.2 Low-Temperature Series 299 
7.8.3 High-Temperature K Expansions 300 
Examples 301 
8. Dimer Assemblies 303 
8.1 Introduction 303 
8.2 The Dimer Partition Function 304 
8.2.1 The Square Lattice Case 306 
8.2.2 The Honeycomb Lattice Case 310 
8.3 The Modified KDP Model Equivalence 317 
8.4 The Ising Model Equivalence 320 
8.5 K-Type and O-Type Transitions 324 
8.6 The Chain Conformal Transition 326 
Examples 332 
A. Appendices 335 
A.1 Fourier Transforms in d Dimensions 335 
A.1.1 Discrete Finite Lattices 335 
A.1.2 A Continuous Finite Volume 336 
A.1.3 A Continuous In,nite Volume 337 
A.1.4 A Special Case 338 
A.2 The Conformal Group 340 
A.3 Group Representation Theory 342 
A.3.1 Groups 342 
A.3.2 Representations 342 
A.3.3 The Block Diagonalization of Transfer Matrices 347 
A.3.4 Equivalence Classes 349 
A.3.5 Using Equivalence Classes for Block Diagonalization 355 
A.3.6 The Transfer Matrix Eigen Problem 357 
A.4 Some Transformations in the Complex Plane 360 
A.5 Algebraic Functions 363 
A.6 Elliptic Integrals, Functions and Nome Series 366 
A.6.1 Elliptic Integrals 366 
A.6.2 Elliptic Functions 367 
A.6.3 Nome Series 368 
A.7 Lattices and Graphs 370 
A.7.1 Subgraphs 370 
A.7.2 Section Graphs 373 
A.7.3 Zero-Field Graphs 374 
A.7.4 Magnetic Graphs and Coefficient Relations 374 
A.7.5 Multi-Bonded Graphs 379 
A.7.6 Polygons and Triangulation 380 
A.7.7 Oriented Graphs 380 
A.8 The Weak-Graph Transformation 381 
A.9 The Generalized Moment-Cumulant Relations 386 
A.10 Kastelyn's Theorem 389 
A.10.1 The Canonical Flux Distribution 389 
A.10.2 The Dimer Partition Function 390 
A.10.3 Superposition Polynomials and Pfaffians 391 
A.11 Determinants of Cyclic Matrices 394 
A.12 The T Matrix 396 
References and Author Index 401 
Subject Index 423 
END
