SBN: 3-540-65321-X
TITLE: Conformal Invariance and Critical Phenomena
AUTHOR: Henkel, Malte
TOC:

1. Critical Phenomena: a Reminder 1
1.1 Phase Diagrams and Critical Exponents 1
1.2 Scale Invariance and Scaling Relations 6
1.3 Some Simple Spin Systems 12
1.3.1 Ising Model 13
1.3.2 Tricritical Ising Model 14
1.3.3 q-States Potts Model 16
1.3.4 Vector Potts Model 19
1.3.5 XY Model 21
1.3.6 YangLee Edge Singularity 23
1.3.7 Percolation 24
1.3.8 Linear Polymers 26
1.3.9 Restricted Solid-On-Solid Models 27
1.4 Some Experimental Examples 30
1.5 Correspondence Between Statistical Systems and Field Theory 37
1.6 Correspondence of Physical Quantities 39
1.6.1 Free Energy Density 40
1.6.2 Correlation Functions 40
1.6.3 Correlation Lengths 41
2. Conformal Invariance 43
2.1 From Scale Invariance to Conformal Invariance 43
2.2 Conformal Transformations in d Dimensions 44
2.3 Conformal Transformations in Two Dimensions 46
2.4 Conformal Invariance in Two Dimensions 49
2.5 Correlation Functions of Quasi-primary Operators 51
2.6 The EnergyMomentum Tensor 53
3. Finite-Size Scaling 63
3.1 Statistical Systems in Finite Geometries 63
3.2 Finite-Size Scaling Hypothesis 64
3.3 Universality 68
3.4 Phenomenological Renormalization 72
3.5 Consequences of Conformal Invariance 74
3.6 Comparison with Experiments 78
4. Representation Theory of the Virasoro Algebra 83
4.1 Verma Module 84
4.2 Hilbert Space Structure 88
4.3 Null Vectors 90
4.4 Kac Formula and Unitarity 92
4.5 Minimal Characters 97
5. Correlators, Null Vectors and Operator Algebra 101
5.1 Null Vectors and Correlation Functions 101
5.2 Operator Algebra and Associativity 104
5.3 Analyticity and the Monodromy Problem 110
5.4 Riemann's Method 112
6. Ising Model Correlators 117
6.1 Spin-Density Four-Point Function 117
6.2 Energy-Density Four-Point Function 121
6.3 Mixed Four-Point Functions 123
6.4 Semi-Local Four-Point Functions 124
7. Coulomb Gas Realization 127
7.1 The Free Bosonic Scalar Field 127
7.2 Screened Coulomb Gas 132
7.3 Minimal Correlation Functions. 134
7.4 Minimal Algebras and OPE Coefficients 137
8. The Hamiltonian Limit and Universality 141
8.1 Hamiltonian Limit in the Ising Model 141
8.2 HubbardStratonovich Transformation 144
8.3 Hamiltonian Limit of the Scalar phi^4 Theory 146
8.4 Hamiltonian Spectrum and Conformal Invariance 148
8.5 TemperleyLieb Algebra 150
8.6 LaudauGinzburg Classification 154
9. Numerical Techniques 157
9.1 Simple Properties of Quantum Hamiltonians 157
9.2 Some Further Physical Quantities and their Critical Exponents 160
9.3 Translation Invariance 162
9.4 Diagonalization 163
9.5 Extrapolation 170
9.5.1 VBS Algorithm 174
9.5.2 BST Algorithm 174
9.6 The DMRG Algorithm 177
10. Conformal Invariance in the Ising Quantum Chain 183
10.1 Exact Diagonalization 183
10.1.1 General Remarks 183
10.1.2 JordanWigner Transformation 184
10.1.3 Diagonalization of a Quadratic Form 185
10.1.4 Eigenvalue Spectrum and Normalization 187
10.2 Character Functions 189
10.3 Finite-Size Scaling Analysis 191
10.3.1 Ground State Energy 191
10.3.2 Operator Content 194
10.3.3 Finite-Size Corrections 197
10.3.4 Finite-Size Scaling Functions 197
10.4 The Spin-1 Quantum Chain 198
10.5 The Virasoro Generators 201
10.6 Recapitulation 203
11. Modular Invariance 205
11.1 The Modular Group 205
11.2 Implementation for Minimal Models 206
11.3 Modular Invariance at c = 1 211
11.3.1 Circle or Coulomb Models 212
11.3.2 Orbifold Models 213
11.4 Lattice Realizations 216
12. Further Developments and Applications 219
12.1 Three-States Potts Model 219
12.2 Tricritical Ising Model 221
12.2.1 Operator Content 221
12.2.2 Supersymmetry and Superconformal Invariance 224
12.3 YangLee Edge Singularity 227
12.4 AshkinTeller Model 230
12.4.1 Relation with the XXZ Quantum Chain 231
12.4.2 Global Symmetry and Boundary Conditions 231
12.4.3 Phase Diagram 233
12.4.4 Operator Content on the c = 1 Line 234
12.5 XY Model 236
12.6 XXZ Quantum Chain 238
12.7 Ising Correlation Functions on Cylinders 242
12.8 Alternative Realizations of the Conformal Algebra 242
12.8.1 Logarithmic Conformal Theories 243
12.8.2 Lattice Two-Point Functions 244
12.9 Percolation 245
12.10 Polymers 247
12.10.1 Linear Polymers 247
12.10.2 Lattice Animals 251
12.11 A Sketch of Conformal Turbulence 254
12.12 Some Remarks on 3D Systems 258
13. Conformal Perturbation Theory 261
13.1 Correlation Functions in the Strip Geometry 261
13.2 General Remarks on Corrections to the Critical Behaviour 263
13.3 Finite-Size Corrections 265
13.3.1 Tower of the Identity 266
13.3.2 Application to the Ising Model 267
13.3.3 Application to the Three-States Potts Model 268
13.3.4 Checking the Operator Content from Finite-Size Corrections 270
13.4 Finite-Size Scaling Functions 270
13.4.1 Ising Model: Thermal Perturbation 271
13.4.2 Ising Model: Magnetic Perturbation 273
13.5 Truncation Method 275
14. The Vicinity of the Critical Point 279
14.1 The c-Theorem 280
14.1.1 Application to Polymers 284
14.2 Conserved Currents Close to Criticality 285
14.3 Exact S-Matrix Approach 289
14.4 Phenomenological Consequences 298
14.4.1 Integrable Perturbations 298
14.4.2 Universal Critical Amplitude Ratios 304
14.4.3 Chiral Potts Model 306
14.4.4 Oriented Interacting Polymers 307
14.4.5 Non-integrable Perturbations 311
14.5 Asymptotic Finite-Size Scaling Functions 316
15. Surface Critical Phenomena 321
15.1 Systems with a Boundary 321
15.2 Conformal Invariance Close to a Free Surface 326
15.3 Finite-Size Scaling with Free Boundary Conditions 330
15.4 Surface Operator Content 332
15.4.1 Ising Model 332
15.4.2 Three-States Potts Model 337
15.4.3 TemperleyLieb Algebra and Relation with the XXZ Chain 338
15.4.4 Tricritical Ising Model 340
15.4.5 YangLee Edge Singularity 340
15.4.6 AshkinTeller Model 340
15.4.7 XXZ Quantum Chain 342
15.4.8 Percolation 343
15.4.9 Polymers 346
15.5 Profiles 346
15.6 Defect Lines 350
15.6.1 Aperiodically Modulated Systems 362
15.6.2 Persistent Currents in Small Rings 363
16. Strongly Anisotropic Scaling 369
16.1 Dynamical Scaling 369
16.2 Schrdinger Invariance 372
16.3 Towards Local Scale Invariance for General theta 377
16.4 Some Remarks on ReactionDiffusion Processes 383
Anhang/Annexe 385
List of Tables 388
List of Figures 390
References 391
Index 411
END
