A primer to the theory of critical phenomena

목차 (Table of Contents)

• CONTENTS
• Preface = xi
• 1. Introduction : Classical Phases and Critical Points
• 1.1 Physical Examples : Macroscopic Properties = 1
• 1.2 Singularity of the Specific Heat = 6
• 1.3 Remarks = 8
• 2. The Ising Model and Its Basic Characteristics in the Mean Field Approximation
• 2.1 The Ising Model and Its Hamiltonian = 9
• 2.2 The Ising Partition Function and Free Energy = 11
• 2.3 Equilibrium Conditions and Thermodynamics = 13
• 2.4 Elementary Consequences of the Model = 15
• 2.5 Discussion : Critical Exponents = 18
• 2.6 The Meaning of the Ising Model and Mean Field Theory = 20
• 2.7 Problem 2.1 : Numerical Checkout = 20
• 2.8 Problem 2.2 : A Physical Estimate of Tc-Ordering Energy vs. Entropy = 22
• Appendix 2.A : Primer in Statistical Physics-A Summary = 22
• 2.A.1 Principles of Thermodynamics (Thermostatics) = 22
• 2.A.2 Connection to Statistical Physics = 24
• 2.A.3 Elementary Example : Statistical Distribution Function for Quantum Multiparticle System : Fermions and Bosons = 26
• 2.A.4 Outlook = 28
• 2.A.5 Final Note = 29
• Appendix 2.B : Expansion of Eq. (2.5) in Powers of M = 29
• 3. General Mean Field Approach
• 3.1 The Heisenberg Model and the Mean Field Approximation = 31
• 3.2 Critical Phenomena in the Mean Field Approximation : A Further Elaboration = 32
• 3.3 Van der Waals Equation of State and Criticality* = 37
• 3.4 Density Fluctuations and Compressibility = 42
• 3.5 Mean Field Theory for Binary Mixtures* = 44
• Appendix 3.A : Useful Relations Related to the van der Waals Equation = 52
• Appendix 3.B : Definition of Correlation Functions = 52
• Appendix 3.C : Gibbs--Duhem Rules = 53
• Appendix 3.D : Formula for the Entropy Part in Eq. (3.61) = 54
• 4. The Landau Theory of Phase Transitions : General Concept and Its Microscopic Relation to Mean Field Theory
• 4.1 Concept of the Order Parameter = 55
• 4.2 Spin Magnetism : Microscopic Derivation of the Landau Theory with Spatial Fluctuations of the Order Parameter* = 67
• 4.2.1 The Concept of Exchange Fields and Ferromagnetism = 67
• 4.2.2 The Appearance of the Gradient Term = 70
• 4.2.3 Formal Equivalence of the Landau Expansion and the Mean-field Approach = 72
• 4.2.4 Landau Theory of the Ferromagnetic Transition = 75
• Supplement: Zero-field Susceptibility in the Ferromagnetic Phase = 77
• 4.2.5 Mean-field Theory for Two-sublattice Antiferromagnets = 77
• 4.3 Rotationally Invariant Form of the Landau Functional = 79
• 4.4 Outlook : Meaning of Mean Field Theory = 80
• Problem 4.1 : A Simple Physical Example of a Continuous Phase Transition: Order-disorder Transformation = 81
• Problem 4.2 : The Entropy and a Continuous Phase Transition for AB Alloys = 83
• 4.5 Historical Note : Order of the Phase Transition = 84
• 5. More General Considerations Concerning Mean Field Theory : The Stratonovich - Hubbard Transformation*
• 5.1 General Form of the Partition Function for the Ising Model = 87
• 5.2 The Upper Critical Dimension = 89
• 5.3 Appendix : Derivation of Eq. (5.4) = 90
• 6. Generalities Relating to the Study of Critical Phenomena
• 6.1 Introduction = 93
• 6.2 Scaling Procedures: Kadanoff Blocks = 94
• 6.3 Operations in Reciprocal Space = 95
• 6.4 Utility of Algebraic Power Laws = 96
• 6.5 Homogeneous Functions = 97
• 6.6 Fixed Points = 98
• 6.7 Appendix : A Simple Example = 99
• 7. Failure of Mean Field Theory and Scaling Methods
• 7.1 Kadanoff Scaling = 101
• 7.2 Properties of the Homogeneous Equation = 103
• 7.3 Scaling Laws for Second Derivatives = 105
• 7.4 Summary and Remarks = 107
• 7.5 Appendix A : Scaling Law for the Chemical Potential = 108
• 8. Kadanoff Scaling
• 8.1 Example Involving a Ring = 111
• 8.2 The Case of Two Dimensions = 115
• 8.3 Rescaling of the Triangular Lattice = 118
• 8.4 Determination of Averaged Perturbation Potentials = 120
• 8.5 Determination of the Fixed Point = 121
• 9. The Renormalization Group Operations
• 9.1 Real Space Renormalization = 123
• 9.2 Renormalization of the Hamiltonian = 124
• 9.3 Tracking Parametric Changes During Coarse Graining = 124
• 9.4 An Example = 125
• 9.5 Croup Properties of the Coarse Graining Operation = 127
• 9.6 Scaling Fields and Properties = 128
• 9.7 Classification of Variables and Related Fixed Points = 128
• 9.8 Back to Homogeneity Relations = 130
• 9.9 Consequences = 132
• Appendix A = 135
• Examples = 135
• 10. Additional Interrelations Between Critical Exponents
• 10.1 Magnetization and Correlation Effects = 139
• 10.2 Effect of Applied Magnetic Fields = 140
• 10.3 Magnetization Close to the Critical Temperature = 141
• 10.4 Magnetic Susceptibility Near the Critical Temperature = 142
• 10.5 Heat Capacity = 142
• 10.6 Summary = 144
• 11. Extension of the Landau Approach to Inhomogeneous Systems and the Physical Picture of Gaussian Fluctuations
• 11.1 Smoothing Operations = 147
• 11.2 Use of Functional Integrals = 149
• Interpretation of the Formalism = 150
• 11.3 Rock-Bottom Approximation = 151
• 11.4 Fourier Transforms and Scaling in Reciprocal Space = 151
• 11.5 Extension : The Ginsburg-Landau Functional in a Rotationally Invariant Case = 153
• 11.6 A Concrete Example : Gaussian Fluctuations and the Ginzburg Criterion = 154
• 12. The Ginzburg-Landau Functional for a Continuous System: Formal Approach to Gaussian Fluctuations
• 12.1 Gaussian Integrals = 159
• 12.2 Partition Function for the Gaussian Model = 161
• 12.3 Operations in k Space = 162
• 12.4 Partition Function = 164
• 12.5 Correlation Functions = 165
• 12.6 Comment = 167
• 12.7 Renormalization of the Hamiltonian = 168
• 12.8 Establishing the Requirements of the Fixed Point = 168
• 12.9 Conclusion = 169
• Appendix 12.A : Hubbard-Stratonovich Transformation = 169
• Appendix 12.B : Relation
• ▽ₓΦ
• ²＝－Φ▽²Φ = 170
• Appendix 12.C : Comment on Functional Integration = 171
• Appendix 12.D : Meaning of Functional (Variational) Derivatives = 172
• 13. The Ginzburg--Landau--Wilson Formalism: Beyond the Gaussian Approximation
• 13.1 Generalities = 175
• 13.2 The RGT Execution = 176
• 13.3 Rescaling Effects = 178
• 13.4 Case d > 4 = 178
• 13.5 Case d < 4 = 180
• 13.6 Determination of the Fixed Point = 181
• 13.7 Case d＝4 = 183
• Appendix 13.A = 184
• 14. Correlation Functions
• 14.1 Correlations Involving Lattice Sites = 185
• 14.2 Correlations in the Continuum Limit = 187
• 14.3 Specification of the Two-Point Connected Correlation Function = 188
• 14.4 Correlations in Direct Space = 190
• Appendix 14.A : Functional Derivatives = 192
• Appendix 14.B : Evaluation of Correlation Function in Direct Space for Three Dimensions = 193
• 15. Beyond the Landau Model
• 15.1 Basics = 195
• 15.2 The Generating Functional for Establishing Correlations = 197
• 15.3 The Two-Point Correlation Function = 199
• 15.4 Postscript = 200
• Appendix 15. A Anomalous Dimension of φ = 200
• 16. An Elementary Examination of Quantum Phase Transitions Involving Fermions*
• 16.1 Introduction : Factors Determining a Continuous Quantum Phase Transition = 203
• 16.2 Example : The Mott-Wigner Criterion for the Electron Gas Instability = 205
• 16.3 Localization on a Lattice : The Hubbard Model = 207
• 16.4 Detailed Discussion of Localization of Fermions : Quantum Critical Points = 209
• 16.5 Quasiparticle Representation of Interacting Electron Systems : From Classical to Quantum Critical Points = 212
• 16.6 Quantum Phase Transition : A Generalized Landau-Hertz Functional = 216
• Formal Remark = 217
• 16.7 An Example = 218
• 16.8 Concluding Remark = 219
• 17. Supplement : Going Beyond the Gaussian Formulation
• 17.1 Method 1 : Two-point Correlation Functions = 221
• 17.2 Method 2 : The Generating Functional for Establishing Correlations for a Single Variable = 224
• 17.2.1 Extension to n Variables = 225
• 17.2.2 Contractions and Wick's Theorem = 226
• 17.2.3 Correlation Functions Involving a Single Variable = 227
• 17.2.4 Determination of Ginzburg-Landau-type Correlation Functions for Several Variables : Brief Introduction to Feynman Diagrams = 228
• 17.2.5 Illustration of the Use of Feynman Diagrams = 230
• 17.2.6 Second-order Expansion = 231
• 17.2.7 Summary = 233
• 17.2.8 Postscript = 233
• 17.2.9 Appendix = 233
• 17.3 Exercises = 234
• Bibliography = 237
• Index = 239