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Conformal Invariance and Critical Phenomena : Theoretical and Mathematical Physics - Malte Henkel

Conformal Invariance and Critical Phenomena

Theoretical and Mathematical Physics


Published: 16th April 1999
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Critical phenomena arise in a wide variety of physical systems. Classi­ cal examples are the liquid-vapour critical point or the paramagnetic­ ferromagnetic transition. Further examples include multicomponent fluids and alloys, superfluids, superconductors, polymers and fully developed tur­ bulence and may even extend to the quark-gluon plasma and the early uni­ verse as a whole. Early theoretical investigators tried to reduce the problem to a very small number of degrees of freedom, such as the van der Waals equation and mean field approximations, culminating in Landau's general theory of critical phenomena. Nowadays, it is understood that the common ground for all these phenomena lies in the presence of strong fluctuations of infinitely many coupled variables. This was made explicit first through the exact solution of the two-dimensional Ising model by Onsager. Systematic subsequent developments have been leading to the scaling theories of critical phenomena and the renormalization group which allow a precise description of the close neighborhood of the critical point, often in good agreement with experiments. In contrast to the general understanding a century ago, the presence of fluctuations on all length scales at a critical point is emphasized today. This can be briefly summarized by saying that at a critical point a system is scale invariant. In addition, conformal invaTiance permits also a non-uniform, local rescal­ ing, provided only that angles remain unchanged.

Critical Phenomena: a Reminderp. 1
Phase Diagrams and Critical Exponentsp. 1
Scale Invariance and Scaling Relationsp. 6
Some Simple Spin Systemsp. 12
Ising Modelp. 13
Tricritical Ising Modelp. 14
q-States Potts Modelp. 16
Vector Potts Modelp. 19
XY Modelp. 21
Yang-Lee Edge Singularityp. 23
Percolationp. 24
Linear Polymersp. 26
Restricted Solid-On-Solid Modelsp. 27
Some Experimental Examplesp. 30
Correspondence Between Statistical Systems and Field Theoryp. 37
Correspondence of Physical Quantitiesp. 39
Free Energy Densityp. 40
Correlation Functionsp. 40
Correlation Lengthsp. 41
Conformal Invariancep. 43
From Scale Invariance to Conformal Invariancep. 43
Conformal Transformations in d Dimensionsp. 44
Conformal Transformations in Two Dimensionsp. 46
Conformal Invariance in Two Dimensionsp. 49
Correlation Functions of Quasi-primary Operatorsp. 51
The Energy-Momentum Tensorp. 53
Finite-Size Scalingp. 63
Statistical Systems in Finite Geometriesp. 63
Finite-Size Scaling Hypothesisp. 64
Universalityp. 68
Phenomenological Renormalizationp. 72
Consequences of Conformal Invariancep. 74
Comparison with Experimentsp. 78
Representation Theory of the Virasoro Algebrap. 83
Verma Modulep. 84
Hilbert Space Structurep. 88
Null Vectorsp. 90
Kac Formula and Unitarityp. 92
Minimal Charactersp. 97
Correlators, Null Vectors and Operator Algebrap. 101
Null Vectors and Correlation Functionsp. 101
Operator Algebra and Associativityp. 104
Analyticity and the Monodromy Problemp. 110
Riemann's Methodp. 112
Ising Model Correlatorsp. 117
Spin-Density Four-Point Functionp. 117
Energy-Density Four-Point Functionp. 121
Mixed Four-Point Functionsp. 123
Semi-Local Four-Point Functionsp. 124
Coulomb Gas Realizationp. 127
The Free Bosonic Scalar Fieldp. 127
Screened Coulomb Gasp. 132
Minimal Correlation Functionsp. 134
Minimal Algebras and OPE Coefficientsp. 137
The Hamiltonian Limit and Universalityp. 141
Hamiltonian Limit in the Ising Modelp. 141
Hubbard-Stratonovich Transformationp. 144
Hamiltonian Limit of the Scalar ¿4 Theoryp. 146
Hamiltonian Spectrum and Conformal Invariancep. 148
Temperley-Lieb Algebrap. 150
Laudau-Ginzburg Classificationp. 154
Numerical Techniquesp. 157
Simple Properties of Quantum Hamiltoniansp. 157
Some Further Physical Quantities and their Critical Exponentsp. 160
Translation Invariancep. 162
Diagonalizationp. 163
Extrapolationp. 170
VBS Algorithmp. 174
BST Algorithmp. 174
The DMRG Algorithmp. 177
Conformal Invariance in the Ising Quantum Chainp. 183
Exact Diagonalizationp. 183
General Remarksp. 183
Jordan-Wigner Transformationp. 184
Diagonalization of a Quadratic Formp. 185
Eigenvalue Spectrum and Normalizationp. 187
Character Functionsp. 189
Finite-Size Scaling Analysisp. 191
Ground State Energyp. 191
Operator Contentp. 194
Finite-Size Correctionsp. 197
Finite-Size Scaling Functionsp. 197
The Spin I Quantum Chainp. 198
The Virasoro Generatorsp. 201
Recapitulationp. 203
Modular Invariancep. 205
The Modular Groupp. 205
Implementation for Minimal Modelsp. 206
Modular Invariance at c =1p. 211
Circle or Coulomb Modelsp. 212
Orbifold Modelsp. 213
Lattice Realizationsp. 216
Further Developments and Applicationsp. 219
Three-States Potts Modelp. 219
Tricritical Ising Modelp. 221
Operator Contentp. 221
Supersymmetry and Superconformal Invariancep. 224
Yang-Lee Edge Singularityp. 227
Ashkin Teller Modelp. 230
Relation with the XXZ Quantum Chainp. 231
Global Symmetry and Boundary Conditionsp. 231
Phase Diagramp. 233
Operator Content on the c =1 Linep. 234
XY Modelp. 236
XXZ Quantum Chainp. 238
Ising Correlation Functions on Cylindersp. 242
Alternative Realizations of the Conformal Algebrap. 242
Logarithmic Conformal Theoriesp. 243
Lattice Two-Point Functionsp. 244
Percolationp. 245
Polymersp. 247
Linear Polymersp. 247
Lattice Animalsp. 251
A Sketch of Conformal Turbulencep. 254
Some Remarks on 3D Systemsp. 258
Conformal Perturbation Theoryp. 261
Correlation Functions in the Strip Geometryp. 261
General Remarks on Corrections to the Critical Behaviourp. 263
Finite-Size Correctionsp. 265
Tower of the Identityp. 266
Application to the Ising Modelp. 267
Application to the Three-States Potts Modelp. 268
Checking the Operator Content from Finite-Size Correctionsp. 270
Finite-Size Scaling Functionsp. 270
Ising Model: Thermal Perturbationp. 271
Ising Model: Magnetic Perturbationp. 273
Truncation Methodp. 275
The Vicinity of the Critical Pointp. 279
The c-Theoremp. 280
Application to Polymersp. 284
Conserved Currents Close to Criticalityp. 285
Exact S-Matrix Approachp. 289
Phenomenological Consequencesp. 298
Integrable Perturbationsp. 298
Universal Critical Amplitude Ratiosp. 304
Chiral Potts Modelp. 306
Oriented Interacting Polymersp. 307
Non-integrable Perturbationsp. 311
Asymptotic Finite-Size Scaling Functionsp. 316
Surface Critical Phenomenap. 321
Systems with a Boundaryp. 321
Conformal Invariance Close to a Free Surfacep. 326
Finite-Size Scaling with Free Boundary Conditionsp. 330
Surface Operator Contentp. 332
Ising Modelp. 332
Three-States Potts Modelp. 337
Temperley-Lieb Algebra and Relation with the XXZ Chainp. 338
Tricritical Ising Modelp. 340
Yang-Lee Edge Singularityp. 340
Ashkin-Teller Modelp. 340
XXZ Quantum Chainp. 342
Percolationp. 343
Polymersp. 346
Profilesp. 346
Defect Linesp. 350
Aperiodically Modulated Systemsp. 362
Persistent Currents in Small Ringsp. 363
Strongly Anisotropic Scalingp. 369
Dynamical Scalingp. 369
Schrödinger Invariancep. 372
Towards Local Scale Invariance for General ¿p. 377
Some Remarks on Reaction-Diffusion Processesp. 383
Anhang/Annexep. 385
List of Tablesp. 388
List of Figuresp. 390
Referencesp. 391
Indexp. 411
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540653219
ISBN-10: 354065321X
Series: Theoretical and Mathematical Physics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 418
Published: 16th April 1999
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 3.81
Weight (kg): 1.75