Conformal Invariance and Critical Phenomena

by Henkel, Malte

• ISBN13: 9783540653219
• ISBN10: 354065321X
• Format: Hardcover
• Copyright: 1999-06-01
• Publisher: Springer Verlag

Summary

This book provides an introduction to conformal field theory and a review of its applications to critical phenomena in condensed-matter systems. After reviewing simple phase transitions and explaining the foundations of conformal invariance and the algebraic methods required, it proceeds to the explicit calculation of four-point correlators. Numerical methods for matrix diagonalization are described as well as finite-size scaling techniques and their conformal extensions. Many exercises are included. Applications treat the Ising, Potts, chiral Potts, Yang-Lee, percolation and XY models, the XXZ chain, linear polymers, tricritical points, conformal turbulence, surface criticality and profiles, defect lines and aperiodically modulated systems, persistent currents and dynamical scaling. The vicinity of the critical point is studied culminating in the exact solution of the two-dimensional Ising model at the critical temperature in a magnetic field. Relevant experimental results are reviewed.

Table of Contents

Critical Phenomena: a Reminder	p. 1
Phase Diagrams and Critical Exponents	p. 1
Scale Invariance and Scaling Relations	p. 6
Some Simple Spin Systems	p. 12
Ising Model	p. 13
Tricritical Ising Model	p. 14
q-States Potts Model	p. 16
Vector Potts Model	p. 19
XY Model	p. 21
Yang-Lee Edge Singularity	p. 23
Percolation	p. 24
Linear Polymers	p. 26
Restricted Solid-On-Solid Models	p. 27
Some Experimental Examples	p. 30
Correspondence Between Statistical Systems and Field Theory	p. 37
Correspondence of Physical Quantities	p. 39
Free Energy Density	p. 40
Correlation Functions	p. 40
Correlation Lengths	p. 41
Conformal Invariance	p. 43
From Scale Invariance to Conformal Invariance	p. 43
Conformal Transformations in d Dimensions	p. 44
Conformal Transformations in Two Dimensions	p. 46
Conformal Invariance in Two Dimensions	p. 49
Correlation Functions of Quasi-primary Operators	p. 51
The Energy-Momentum Tensor	p. 53
Finite-Size Scaling	p. 63
Statistical Systems in Finite Geometries	p. 63
Finite-Size Scaling Hypothesis	p. 64
Universality	p. 68
Phenomenological Renormalization	p. 72
Consequences of Conformal Invariance	p. 74
Comparison with Experiments	p. 78
Representation Theory of the Virasoro Algebra	p. 83
Verma Module	p. 84
Hilbert Space Structure	p. 88
Null Vectors	p. 90
Kac Formula and Unitarity	p. 92
Minimal Characters	p. 97
Correlators, Null Vectors and Operator Algebra	p. 101
Null Vectors and Correlation Functions	p. 101
Operator Algebra and Associativity	p. 104
Analyticity and the Monodromy Problem	p. 110
Riemann's Method	p. 112
Ising Model Correlators	p. 117
Spin-Density Four-Point Function	p. 117
Energy-Density Four-Point Function	p. 121
Mixed Four-Point Functions	p. 123
Semi-Local Four-Point Functions	p. 124
Coulomb Gas Realization	p. 127
The Free Bosonic Scalar Field	p. 127
Screened Coulomb Gas	p. 132
Minimal Correlation Functions	p. 134
Minimal Algebras and OPE Coefficients	p. 137
The Hamiltonian Limit and Universality	p. 141
Hamiltonian Limit in the Ising Model	p. 141
Hubbard-Stratonovich Transformation	p. 144
Hamiltonian Limit of the Scalar ¿4 Theory	p. 146
Hamiltonian Spectrum and Conformal Invariance	p. 148
Temperley-Lieb Algebra	p. 150
Laudau-Ginzburg Classification	p. 154
Numerical Techniques	p. 157
Simple Properties of Quantum Hamiltonians	p. 157
Some Further Physical Quantities and their Critical Exponents	p. 160
Translation Invariance	p. 162
Diagonalization	p. 163
Extrapolation	p. 170
VBS Algorithm	p. 174
BST Algorithm	p. 174
The DMRG Algorithm	p. 177
Conformal Invariance in the Ising Quantum Chain	p. 183
Exact Diagonalization	p. 183
General Remarks	p. 183
Jordan-Wigner Transformation	p. 184
Diagonalization of a Quadratic Form	p. 185
Eigenvalue Spectrum and Normalization	p. 187
Character Functions	p. 189
Finite-Size Scaling Analysis	p. 191
Ground State Energy	p. 191
Operator Content	p. 194
Finite-Size Corrections	p. 197
Finite-Size Scaling Functions	p. 197
The Spin I Quantum Chain	p. 198
The Virasoro Generators	p. 201
Recapitulation	p. 203
Modular Invariance	p. 205
The Modular Group	p. 205
Implementation for Minimal Models	p. 206
Modular Invariance at c =1	p. 211
Circle or Coulomb Models	p. 212
Orbifold Models	p. 213
Lattice Realizations	p. 216
Further Developments and Applications	p. 219
Three-States Potts Model	p. 219
Tricritical Ising Model	p. 221
Operator Content	p. 221
Supersymmetry and Superconformal Invariance	p. 224
Yang-Lee Edge Singularity	p. 227
Ashkin Teller Model	p. 230
Relation with the XXZ Quantum Chain	p. 231
Global Symmetry and Boundary Conditions	p. 231
Phase Diagram	p. 233
Operator Content on the c =1 Line	p. 234
XY Model	p. 236
XXZ Quantum Chain	p. 238
Ising Correlation Functions on Cylinders	p. 242
Alternative Realizations of the Conformal Algebra	p. 242
Logarithmic Conformal Theories	p. 243
Lattice Two-Point Functions	p. 244
Percolation	p. 245
Polymers	p. 247
Linear Polymers	p. 247
Lattice Animals	p. 251
A Sketch of Conformal Turbulence	p. 254
Some Remarks on 3D Systems	p. 258
Conformal Perturbation Theory	p. 261
Correlation Functions in the Strip Geometry	p. 261
General Remarks on Corrections to the Critical Behaviour	p. 263
Finite-Size Corrections	p. 265
Tower of the Identity	p. 266
Application to the Ising Model	p. 267
Application to the Three-States Potts Model	p. 268
Checking the Operator Content from Finite-Size Corrections	p. 270
Finite-Size Scaling Functions	p. 270
Ising Model: Thermal Perturbation	p. 271
Ising Model: Magnetic Perturbation	p. 273
Truncation Method	p. 275
The Vicinity of the Critical Point	p. 279
The c-Theorem	p. 280
Application to Polymers	p. 284
Conserved Currents Close to Criticality	p. 285
Exact S-Matrix Approach	p. 289
Phenomenological Consequences	p. 298
Integrable Perturbations	p. 298
Universal Critical Amplitude Ratios	p. 304
Chiral Potts Model	p. 306
Oriented Interacting Polymers	p. 307
Non-integrable Perturbations	p. 311
Asymptotic Finite-Size Scaling Functions	p. 316
Surface Critical Phenomena	p. 321
Systems with a Boundary	p. 321
Conformal Invariance Close to a Free Surface	p. 326
Finite-Size Scaling with Free Boundary Conditions	p. 330
Surface Operator Content	p. 332
Ising Model	p. 332
Three-States Potts Model	p. 337
Temperley-Lieb Algebra and Relation with the XXZ Chain	p. 338
Tricritical Ising Model	p. 340
Yang-Lee Edge Singularity	p. 340
Ashkin-Teller Model	p. 340
XXZ Quantum Chain	p. 342
Percolation	p. 343
Polymers	p. 346
Profiles	p. 346
Defect Lines	p. 350
Aperiodically Modulated Systems	p. 362
Persistent Currents in Small Rings	p. 363
Strongly Anisotropic Scaling	p. 369
Dynamical Scaling	p. 369
Schrödinger Invariance	p. 372
Towards Local Scale Invariance for General ¿	p. 377
Some Remarks on Reaction-Diffusion Processes	p. 383
Anhang/Annexe	p. 385
List of Tables	p. 388
List of Figures	p. 390
References	p. 391
Index	p. 411
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