Introduction to conformal invariance and its applications to critical phenomena (LNPm016, Springer 1993)

Cover ......Page 1
Lecture Notes in Physics: Monographs 16 ......Page 2
Introduction to Conformal Invariance and Its Applications to Critical Phenomena ......Page 4
Preface ......Page 6
Contents ......Page 10
List of tables ......Page 14
List of figures ......Page 15
1.1 Phase diagrams and critical exponents ......Page 18
1.2 Scaling relations ......Page 21
1.3 Some simple spin systems ......Page 24
The Yang-Lee edge singularity ......Page 25
The tricritical Ising model ......Page 26
The p-state Potts model ......Page 27
The vector Potts model ......Page 31
The XY model ......Page 32
Restricted solid-on-solid models ......Page 33
1.4 Correspondence between statistical systems and field theory ......Page 35
Free energy density ......Page 38
Correlation lengths ......Page 39
2.1 Conformal group ......Page 41
2.2 Conformal algebra in two dimensions ......Page 42
2.3 Conformal theory in two dimensions ......Page 44
2.4 Elementary correlation functions ......Page 45
2.5 The stress-energy tensor ......Page 47
3.1 Statistical systems in finite geometries ......Page 55
3.2 Finite-size scaling hypothesis ......Page 56
3.3 Universality ......Page 59
3.4 Phenomenological renormalization ......Page 62
3.5 Consequences of conformal invariance ......Page 64
4.1 Verma module ......Page 68
4.2 Hilbert space structure ......Page 71
4.3 Null-vectors ......Page 73
4.4 Null-vectors and correlation functions ......Page 74
4.5 Kac formula and unitarity ......Page 77
4.6 Minimal characters ......Page 79
5.1 Operator algebra and associativity ......Page 82
5.2 Analyticity and the monodromy problem ......Page 87
5.3 The Riemann point of view ......Page 89
6.1 Four-point function of the spin-density operator ......Page 93
6.2 Four-point function of the energy-density operator ......Page 96
6.3 Mixed four-point function ......Page 98
6.4 Semi-local four-point functions ......Page 99
7.1 Free massless boson gas ......Page 101
7.2 Screened Coulomb gas ......Page 104
7.3 Minimal correlation functions ......Page 106
7.4 Minimal algebras and OPE coefficients ......Page 108
8.1 Hamiltonian limit in the Ising model ......Page 111
8.2 Hubbard-Stratovoic transformation ......Page 114
8.3 Hamiltonian limit of the scalar phi^4 theory ......Page 115
8.5 Temperley-Lieb algebra ......Page 118
8.6 Landan-Ginzburg classification ......Page 122
9.1 Simple properties of quantum Hamiltonians ......Page 124
9.2 Some further physical quantities and their critical exponents ......Page 126
9.3 Translation invariance ......Page 128
9.4 Diagonalization ......Page 129
9.5 Extrapolation ......Page 132
VBS algorithm ......Page 135
BST algorithm ......Page 136
General remarks ......Page 139
Jordan-Wigner transformation ......Page 140
Diagonalization of a quadratic form ......Page 141
Eigenvalue spectrum and normalization ......Page 142
10.2 Character functions ......Page 143
Ground state energy ......Page 145
Operator content ......Page 147
Finite-size corrections ......Page 149
Finite-size scaling functions ......Page 150
10.4 The Virasoro generators ......Page 151
10.5 Recapitulation ......Page 152
11.2 Implementation for minimal models ......Page 154
Circle or Coulomb models ......Page 160
Orbifold models ......Page 162
12.1 Three-states Potts model ......Page 165
Operator content ......Page 166
Supersymmetry and superconformal invariance ......Page 169
12.3 Yang-Lee edge singularity ......Page 171
12.4 Ashkin-Teller model ......Page 174
Global symmetry and boundary conditions ......Page 175
Phase diagram ......Page 176
Operator content on the c = 1 line ......Page 177
12.5 XY model ......Page 179
12.6 XXZ quantum chain ......Page 181
12.7 A few remarks on 3D systems ......Page 185
13.1 Correlation functions in the strip geometry ......Page 188
13.2 General remarks on corrections to the critical behaviour ......Page 189
13.3 Finite-size Corrections ......Page 191
Application to the Ising model ......Page 193
Application to the three-states Potts model ......Page 194
13.4 Finite-size scaling functions ......Page 196
Ising model: thermal perturbation ......Page 197
Ising model: magnetic perturbation ......Page 198
13.5 Truncation method ......Page 200
14.1 The c-theorem ......Page 204
14.2 Conserved currents close to criticality ......Page 207
14.3 Exact S-matrix approach ......Page 210
Integrable perturbations ......Page 217
Non-integrable perturbations ......Page 220
14.5 Thermodynamic Bethe Ansatz ......Page 224
14.6 Asymptotic finite-size scaling functions ......Page 230
15.1 Systems with a boundary ......Page 234
15.2 Conformal invariance close to a free surface ......Page 238
15.3 Finite-size scaling with free boundary conditions ......Page 241
Ising model ......Page 242
Three-states Ports model ......Page 245
Yang-Lee edge singularity ......Page 247
Ashkin-Teller model ......Page 248
XXZ quantum chain ......Page 249
15.5 Defect lines ......Page 250
16 Outlook: Beyond the Conformal Group ......Page 259
References ......Page 264
Index ......Page 273