Physics of Critical Fluctuations

1. Classical Approach.- 1.1 Introduction.- 1.2 Landau Theory.- 1.2.1 The Scalar Order Parameter.- 1.2.2 The Vector Order Parameter.- 1.3 Broken Symmetry and Condensation.- 1.3.1 Bose-Einstein Condensation.- 1.4 Ergodicity.- 1.4.1 Susceptibility.- 1.4.2 The Ergodic Hypothesis.- 1.5 Gaussian Approximation.- 1.5.1 Goldstone Branch of Excitations.- 1.5.2 Correlation Functions.- 1.5.3 Microscopic Scales in Phase Transitions.- 1.6 The Ginzburg Criterion.- 1.6.1 Critical Dimensions.- 1.7 The Scaling Hypothesis.- 1.7.1 Scaling Laws.- 2 The Ginzburg-Landau Functional.- 2.1 Introduction.- 2.2 Classical Systems.- 2.2.1 The Ising Model.- 2.2.2 The Heisenberg Model.- 2.2.3 Interacting Particles.- 2.3 Quantum Systems.- 2.3.1 The Heisenberg Hamiltonian.- 2.3.2 Bose Gas.- 2.3.3 Bose-Einstein Condensation.- 2.3.4 Fermi Gas.- 3 Wilson’s Renormalization Scheme.- 3.1 Introduction.- 3.2 Kadanoff’s Invariance.- 3.3 Wilson’s Theory.- 3.3.1 Derivation of the RG Equation.- 3.3.2 Linearized RG Equation.- 3.3.3 Redundant Eigenvectors.- 3.3.4 Scaling Properties and Critical Exponents.- 3.3.5 Gaussian Fixed Point.- 3.3.6 The Scaling-Field Method.- 3.3.7 ?-Expansion in Scaling Fields.- 4 Field Theoretical RG.- 4.1 Introduction.- 4.2 Perturbation Theory.- 4.2.1 General Definitions.- 4.2.2 Graph Equations for Ø4-Model.- 4.2.3 Rules of Evaluation.- 4.2.4 Regularization.- 4.3 Renormalization.- 4.3.1 Calculus of Divergences.- 4.3.2 Simple Generalizations.- 4.3.3 Multiplicative Group Equations.- 4.3.4 Scaling Properties.- 4.3.5 Differential Group Equations.- 4.4 Scaling Laws.- 4.5 ?-Expansion.- 4.5.1 Addition of Counter-Terms.- 4.5.2 An Alternative Approach.- 4.5.3 Results.- 4.5.4 Correction to Scaling.- 4.5.5 Improvement of Convergence.- 4.6 Expansions at d = 3.- 4.6.1 Comparison of Massive and Massless Theories.- 4.6.2 Experimental Situation.- 4.7 Results of Different Approaches.- 5 Generalized RG Approach.- 5.1 Introduction.- 5.2 Scale Transformations.- 5.2.1 Differential Form of Scale Covariance.- 5.2.2 Two Ways of Utilizing Scale Covariance.- 5.3 Scale Equations.- 5.3.1 Conventional Approach.- 5.3.2 Generalized Approach.- 5.4 RG Equations.- 5.4.1 Conventional Approach.- 5.4.2 Generalized Approach.- 5.4.3 RG as a Characteristic Set of Scale Equations.- 5.5 Applications of Generalized SE.- 5.5.1 Structure of the Correlation Function at the Transition Point.- 5.5.2 Function ?(q) at the Fixed Point.- 5.6 ?-Expansion.- 5.6.1 General Scheme of the Method.- 5.6.2 Solution of Equations.- 5.6.3 Evaluation of the ?-Exponent.- 5.6.4 Evaluation of the v-Exponent.- 5.7 Comparison of Different RG Approaches.- 6 RG Study of Particular Systems.- 6.1 Reduction of the Characteristic Set.- 6.2 The Gaussian Model.- 6.3 The Ø4 Model.- 6.3.1 General Consideration.- 6.3.2 ð(n) Symmetry.- 6.3.3 Cubic Symmetry.- 6.3.4 Interacting Fields.- 6.3.5 Logarithmical Corrections.- 6.4 The Ø4+Ø6-Model.- 6.5 Free Energy in the Critical Region.- 6.5.1 Quatric Form in the Instability Region.- 6.5.2 Cubic Symmetry.- 6.5.3 Interacting Fields.- 7 Competing Interactions.- 7.1 Heisenberg Magnets.- 7.1.1 RG Equations.- 7.1.2 Fixed Points.- 7.1.3 Flow Lines and Phase Diagrams.- 7.2 Bicritical Points in Antiferromagnets.- 7.2.1 Ginzburg-Landau Functional.- 7.2.2 RG Analysis.- 7.2.3 Experimental Data.- 7.3 Dipole Interaction.- 7.3.1 Dipole Hamiltonian.- 7.3.2 RG Analysis.- 7.3.3 Cubic Symmetry.- 7.3.4 Tetragonal Symmetry.- 7.4 Impure Systems.- 7.4.1 Harris’s Criterion.- 7.4.2 The Replica Method.- 7.4.3 RG Analysis.- 7.4.4 Flow Line Runaway.- 7.4.5 Loop Renormalizations.- 7.4.6 Anisotropic Systems.- 7.4.7 Systems with Competing Interactions.- 8 Exactly Solvable Models and RG.- 8.1 Fluctuation Effects in Spherical Model.- 8.1.1 Inhomogeneous Ordering.- 8.1.2 Homogeneous Ordering.- 8.2 Inversion of Phase Transitions.- 8.3 Model with Reduced Interaction.- 8.3.1 General Consideration.- 8.3.2 Critical Exponents. Crossover.- 8.4 First-Order Transitions.- 8.4.1 Cubic Symmetry.- 8.4.2 Interacting Fields.- 8.5 RG and Reduced Interactions.- 8.5.1 The RG Equation for the Model.- 8.5.2 Critical Exponents.- 9 Application to Copper Oxides.- 9.1 Introduction.- 9.2 La2CuO4 Systems.- 9.2.1 The Ginzburg-Landau Functional.- 9.2.2 Mean Field Approximation.- 9.2.3 RG Analysis.- 9.2.4 The Influence of Impurities.- 9.3 Oxygen Ordering in ABa2Cu3O6+x.- 9.3.1 Description of the Model.- 9.3.2 Free Energy.- 9.3.3 Phase Transition.- 9.3.4 Change of Oxygen Concentration.- 9.4 d-Pairing in the Superconducting State.- 9.4.1 RG Approach.- 9.4.2 An Exactly Solvable Model.- 9.4.3 Comparison with RG.- 9.4.4 Conclusion.- A Evaluation of Integrals.- A.1 Gaussian Integral.- A.2 A Typical Diagram.- B Local RG.- B.1 General Consideration.- B.2 Spectral Theorems.