Exactly Solved Models in Statistical Mechanics

by Baxter, Rodney J.

• ISBN13: 9780486462714
• ISBN10: 0486462714
• Format: Paperback
• Copyright: 2008-01-11
• Publisher: Dover Publications

Summary

This text explores two-dimensional lattice models in statistical mechanics and illustrates methods for their solution. Comprehensive but concise, it indicates the routes between equations without superfluous details. Author R. J. Baxter is a fellow of the Royal Society of London and the Australian Academy of Science, as well as Emeritus Professor of the Mathematical Sciences Institute at Australian National University, Canberra. Professor Baxter has updated this edition with a new chapter covering recent developments.

Author Biography

Rodney J. Baxter is Professor Emeritus at Australian National University, Canberra.

Table of Contents

Preface	p. v
Basic Statistical Mechanics
Phase transitions and critical points	p. 1
The scaling hypothesis	p. 4
Universality	p. 7
The partition function	p. 8
Approximation methods	p. 9
Exactly solved models	p. 11
The general Ising model	p. 14
Nearest-neighbour Ising model	p. 21
The lattice gas	p. 24
The van der Waals fluid and classical exponents	p. 30
The One-dimensional Ising Model
Free energy and magnetization	p. 32
Correlations	p. 35
Critical behaviour near T = 0	p. 37
The Mean Field Model
Thermodynamic properties	p. 39
Phase transition	p. 42
Zero-field properties and critical exponents	p. 44
Critical equation of state	p. 45
Mean field lattice gas	p. 46
Ising Model on the Bethe Lattice
The Bethe lattice	p. 47
Dimensionality	p. 49
Recurrence relations for the central magnetization	p. 49
The limit n to [infinity]	p. 51
Magnetization as a function of H	p. 53
Free energy	p. 55
Low-temperature zero-field results	p. 56
Critical behaviour	p. 57
Anisotropic model	p. 58
The Spherical Model
Formulation of the model	p. 60
Free energy	p. 61
Equation of state and internal energy	p. 64
The function g'(z)	p. 65
Existence of a critical point for d > 2	p. 66
Zero-field properties: exponents [alpha], [beta], [gamma], [gamma]'	p. 68
Critical equation of state	p. 70
Duality and Star-Triangle Transformations of Planar Ising Models
General comments on two-dimensional models	p. 72
Duality relation for the square lattice Ising model	p. 73
Honeycomb-triangular duality	p. 78
Star-triangle relation	p. 80
Triangular-triangular duality	p. 86
Square-Lattice Ising Model
Historical introduction	p. 88
The transfer matrices V, W	p. 89
Two significant properties of V and W	p. 91
Symmetry relations	p. 95
Commutation relations for transfer matrices	p. 96
Functional relation for the eigenvalues	p. 97
Eigenvalues [Lambda] for T = T[subscript c]	p. 98
Eigenvalues [Lambda] for T < T[subscript c]	p. 101
General expressions for the eigenvalues	p. 108
Next-largest eigenvalues: interfacial tension, correlation length and magnetization for T < T[subscript c]	p. 111
Next-largest eigenvalue and correlation length for T > T[subscript c]	p. 119
Critical behaviour	p. 120
Parametrized star-triangle relation	p. 122
The dimer problem	p. 124
Ice-Type Models
Introduction	p. 127
The transfer matrix	p. 130
Line-conservation	p. 131
Eigenvalues for arbitrary n	p. 138
Maximum Eigenvalue: location of z[subscript 1], ..., z[subscript n]	p. 140
The case [Delta] > 1	p. 143
Thermodynamic limit for [Delta] < 1	p. 143
Free energy for - 1 < [Delta] < 1	p. 145
Free energy for [Delta] < -1	p. 148
Classification of phases	p. 150
Critical singularities	p. 156
Ferroelectric model in a field	p. 160
Three-colourings of the square lattice	p. 165
Alternative Way of Solving the Ice-Type Models
Introduction	p. 180
Commuting transfer matrices	p. 180
Equations for the eigenvalues	p. 181
Matrix function relation that defines the eigenvalues	p. 182
Summary of the relevant matrix properties	p. 184
Direct derivation of the matrix properties: commutation	p. 185
Parametrization in terms of entire functions	p. 190
The matrix Q([upsilon])	p. 192
Values of [rho], [lambda], [upsilon]	p. 200
Square Lattice Eight-Vertex Model
Introduction	p. 202
Symmetries	p. 204
Formulation as an Ising model with two- and four-spin interactions	p. 207
Star - triangle relation	p. 210
The matrix Q([upsilon])	p. 215
Equations for the eigenvalues of V([upsilon])	p. 222
Maximum eigenvalue: location of [upsilon subscript 1], ...,[upsilon subscript n]	p. 224
Calculation of the free energy	p. 228
The Ising case	p. 237
Other thermodynamic properties	p. 239
Classification of phases	p. 245
Critical singularities	p. 248
An equivalent Ising model	p. 255
The XYZ chain	p. 258
Summary of definitions of [Delta], [Gamma], k, [lambda], [upsilon], q, x, z, p, [mu], w	p. 267
Special cases	p. 269
An exactly solvable inhomogeneous eight-vertex model	p. 272
Kagome Lattice Eight-Vertex Model
Definition of the model	p. 276
Conversion to a square-lattice model	p. 281
Correlation length and spontaneous polarization	p. 284
Free energy	p. 285
Formulation as a triangular-honeycomb Ising model with two- and four-spin interactions	p. 286
Phases	p. 293
K" = 0: The triangular and honeycomb Ising models	p. 294
Explicit expansions of the Ising model results	p. 300
Thirty-two vertex model	p. 309
Triangular three-spin model	p. 314
Potts and Ashkin-Teller Models
Introduction and definition of the Potts model	p. 322
Potts model and the dichromatic polynomial	p. 323
Planar graphs: equivalent ice-type model	p. 325
Square-lattice Potts model	p. 332
Critical square-lattice Potts model	p. 339
Triangular-lattice Potts model	p. 345
Combined formulae for all three planar lattice Potts models	p. 350
Critical exponents of the two-dimensional Potts model	p. 351
Square-lattice Ashkin-Teller model	p. 353
Corner Transfer Matrices
Definitions	p. 363
Expressions as products of operators	p. 369
Star-triangle relation	p. 370
The infinite lattice limit	p. 376
Eigenvalues of the CTMs	p. 377
Inversion properties: relation for [kappa](u)	p. 382
Eight-vertex model	p. 385
Equations for the CTMs	p. 389
Hard Hexagon and Related Models
Historical background and principal results	p. 402
Hard square model with diagonal interactions	p. 409
Free energy	p. 420
Sub-lattice densities and the order parameter R	p. 426
Explicit formulae for the various cases: the Rogers-Ramanujan identities	p. 432
Alternative expressions for the [kappa], [rho], R	p. 443
The hard hexagon model	p. 448
Comments and speculations	p. 452
Acknowledgements	p. 454
Elliptic Functions
Definitions	p. 455
Analyticity and periodicity	p. 456
General theorems	p. 458
Algebraic identities	p. 460
Differential and integral identities	p. 464
Landen transformation	p. 466
Conjugate modulus	p. 467
Poisson summation formula	p. 468
Series expansions of the theta functions	p. 469
Parametrization of symmetric biquadratic relations	p. 471
Subsequent Developments
Introduction	p. 474
Three-dimensional models	p. 474
Chiral Potts model	p. 475
References	p. 485
Supplementary References	p. 493
Index	p. 495
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