Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics

Contents......Page 18
Part I: Preliminary Notions......Page 24
1.1 Phase Transitions......Page 26
1.2 The Ising Model......Page 41
1A: Ensembles in Classical Statistical Mechanics......Page 44
1B: Ensembles in Quantum Statistical Mechanics......Page 49
Problems......Page 61
2.1 Recursive Approach......Page 68
2.2 Transfer Matrix......Page 74
2.3 Series Expansions......Page 82
2.4 Critical Exponents and Scaling Laws......Page 84
2.5 The Potts Model......Page 85
2.6 Models with O(n) Symmetry......Page 90
2.7 Models with Z[sub(n)] Symmetry......Page 97
2.8 Feynman Gas......Page 100
2A: Special Functions......Page 101
2B: n-dimensional Solid Angle......Page 108
2C: The Four-color Problem......Page 109
Problems......Page 117
3.1 Mean Field Theory of the Ising Model......Page 120
3.2 Mean Field Theory of the Potts Model......Page 125
3.3 Bethe–Peierls Approximation......Page 128
3.4 The Gaussian Model......Page 132
3.5 The Spherical Model......Page 141
3A: The Saddle Point Method......Page 148
3B: Brownian Motion on a Lattice......Page 151
Problems......Page 163
Part II: Bidimensional Lattice Models......Page 168
4 Duality of the Two-dimensional Ising Model......Page 170
4.1 Peierls’s Argument......Page 171
4.2 Duality Relation in Square Lattices......Page 172
4.3 Duality Relation between Hexagonal and Triangular Lattices......Page 178
4.4 Star–Triangle Identity......Page 180
4.5 Critical Temperature of Ising Model in Triangle and Hexagonal Lattices......Page 182
4.6 Duality in Two Dimensions......Page 184
4A: Numerical Series......Page 190
4B: Poisson Resummation Formula......Page 191
Problems......Page 193
5.1 Combinatorial Approach......Page 195
5.2 Dimer Method......Page 205
Problems......Page 214
6 Transfer Matrix of the Two-dimensional Ising Model......Page 215
6.1 Baxter’s Approach......Page 216
6.2 Eigenvalue Spectrum at the Critical Point......Page 226
6.4 Yang–Baxter Equation and R-matrix......Page 229
Problems......Page 234
Part III: Quantum Field Theory and Conformal Invariance......Page 238
7.1 Motivations......Page 240
7.2 Order Parameters and Lagrangian......Page 242
7.3 Field Theory of the Ising Model......Page 246
7.4 Correlation Functions and Propagator......Page 248
7.5 Perturbation Theory and Feynman Diagrams......Page 251
7.6 Legendre Transformation and Vertex Functions......Page 257
7.7 Spontaneous Symmetry Breaking and Multicriticality......Page 260
7.8 Renormalization......Page 264
7.9 Field Theory in Minkowski Space......Page 268
7.10 Particles......Page 272
7.11 Correlation Functions and Scattering Processes......Page 275
7A: Feynman Path Integral Formulation......Page 277
7B: Relativistic Invariance......Page 279
7C: Noether’s Theorem......Page 281
Problems......Page 283
8.1 Introduction......Page 287
8.2 Reducing the Degrees of Freedom......Page 289
8.3 Transformation Laws and Effective Hamiltonians......Page 290
8.4 Fixed Points......Page 294
8.5 The Ising Model......Page 296
8.6 The Gaussian Model......Page 300
8.7 Operators and Quantum Field Theory......Page 301
8.8 Functional Form of the Free Energy......Page 303
8.9 Critical Exponents and Universal Ratios......Page 305
8.10 Β-functions......Page 308
Problems......Page 311
9.1 Introduction......Page 313
9.2 Transfer Matrix and Hamiltonian Limit......Page 314
9.3 Order and Disorder Operators......Page 318
9.4 Perturbation Theory......Page 320
9.5 Expectation Values of Order and Disorder Operators......Page 322
9.6 Diagonalization of the Hamiltonian......Page 323
9.7 Dirac Equation......Page 328
Problems......Page 331
10.1 Introduction......Page 333
10.2 The Algebra of Local Fields......Page 334
10.3 Conformal Invariance......Page 338
10.4 Quasi–Primary Fields......Page 341
10.5 Two-dimensional Conformal Transformations......Page 343
10.6 Ward Identity and Primary Fields......Page 348
10.7 Central Charge and Virasoro Algebra......Page 352
10.8 Representation Theory......Page 358
10.9 Hamiltonian on a Cylinder Geometry and the Casimir Effect......Page 367
10A: Moebius Transformations......Page 370
Problems......Page 377
11.2 Null Vectors and Kac Determinant......Page 381
11.3 Unitary Representations......Page 385
11.4 Minimal Models......Page 386
11.5 Coulomb Gas......Page 393
11.6 Landau–Ginzburg Formulation......Page 405
11.7 Modular Invariance......Page 408
11A: Hypergeometric Functions......Page 416
Problems......Page 418
12.2 Conformal Field Theory of a Free Bosonic Field......Page 420
12.3 Conformal Field Theory of a Free Fermionic Field......Page 431
12.4 Bosonization......Page 442
Problems......Page 445
13.2 Superconformal Models......Page 449
13.3 Parafermion Models......Page 454
13.4 Kac–Moody Algebra......Page 461
13.5 Conformal Models as Cosets......Page 471
13A: Lie Algebra......Page 475
Problems......Page 485
14.2 The Ising Model......Page 487
14.3 The Universality Class of the Tricritical Ising Model......Page 498
14.4 Three-state Potts Model......Page 501
14.5 The Yang–Lee Model......Page 504
14.6 Conformal Models with O(n) Symmetry......Page 507
Problems......Page 509
Part IV: Away from Criticality......Page 510
15.1 Introduction......Page 512
15.2 Conformal Perturbation Theory......Page 514
15.3 Example: The Two-point Function of the Yang–Lee Model......Page 520
15.4 Renormalization Group and Β-functions......Page 522
15.5 C-theorem......Page 527
15.6 Applications of the c-theorem......Page 530
15.7 Δ-theorem......Page 535
16.1 Introduction......Page 539
16.2 The Sinh–Gordon Model......Page 540
16.3 The Sine–Gordon Model......Page 546
16.4 The Bullogh–Dodd Model......Page 550
16.5 Integrability versus Non-integrability......Page 553
16.6 The Toda Field Theories......Page 555
16.7 Toda Field Theories with Imaginary Coupling Constant......Page 565
16.8 Deformation of Conformal Conservation Laws......Page 566
16.9 Multiple Deformations of Conformal Field Theories......Page 574
Problems......Page 578
17 S-Matrix Theory......Page 580
17.1 Analytic Scattering Theory......Page 581
17.2 General Properties of Purely Elastic Scattering Matrices......Page 591
17.3 Unitarity and Crossing Invariance Equations......Page 597
17.4 Analytic Structure and Bootstrap Equations......Page 602
17.5 Conserved Charges and Consistency Equations......Page 606
17A: Historical Development of S-Matrix Theory......Page 610
17B: Scattering Processes in Quantum Mechanics......Page 613
17C: n-particle Phase Space......Page 618
Problems......Page 624
18.1 Yang–Lee and Bullogh–Dodd Models......Page 628
18.2 φ[sub(1,3)] Integrable Deformation of the Conformal Minimal Models M[sub(2,2n+3)]......Page 631
18.3 Multiple Poles......Page 634
18.4 S-Matrices of the Ising Model......Page 635
18.5 The Tricritical Ising Model at T≠Tc......Page 642
18.6 Thermal Deformation of the Three-state Potts Model......Page 646
18.7 Models with Internal O(n) Invariance......Page 649
18.8 S-Matrix of the Sine–Gordon Model......Page 654
18.9 S-Matrices for φ[sub(1,3)], φ[sub(1,2)], φ[sub(2,1) Deformation of Minimal Models......Page 658
Problems......Page 674
19.2 Casimir Energy......Page 678
19.3 Bethe Relativistic Wave Function......Page 681
19.4 Derivation of Thermodynamics......Page 683
19.5 The Meaning of the Pseudo-energy......Page 688
19.6 Infrared and Ultraviolet Limits......Page 691
19.7 The Coefficient of the Bulk Energy......Page 694
19.8 The General Form of the TBA Equations......Page 695
19.9 The Exact Relation λ(m)......Page 698
19.10 Examples......Page 700
19.11 Thermodynamics of the Free Field Theories......Page 703
19.12 L-channel Quantization......Page 705
Problems......Page 711
20 Form Factors and Correlation Functions......Page 712
20.1 General Properties of the Form Factors......Page 713
20.2 Watson’s Equations......Page 715
20.3 Recursive Equations......Page 718
20.5 Correlation Functions......Page 720
20.6 Form Factors of the Stress–Energy Tensor......Page 724
20.7 Vacuum Expectation Values......Page 726
20.8 Ultraviolet Limit......Page 729
20.9 The Ising Model at T≠Tc......Page 732
20.10Form Factors of the Sinh–Gordon Model......Page 737
20.11The Ising Model in a Magnetic Field......Page 743
Problems......Page 748
21.1 Multiple Deformations of the Conformal Field Theories......Page 751
21.2 Form Factor Perturbation Theory......Page 753
21.3 First-order Perturbation Theory......Page 757
21.4 Non-locality and Confinement......Page 761
21.5 The Scaling Region of the Ising Model......Page 762
Problems......Page 768
B......Page 770
C......Page 771
D......Page 772
G......Page 773
M......Page 774
P......Page 775
S......Page 776
T......Page 777
Z......Page 778