A Primer to the Theory of Critical Phenomena

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A Primer to the Theory of Critical Phenomena Front-Matter_2018_A-Primer-to-the-Theory-of-Critical-Phenomena Copyright_2018_A-Primer-to-the-Theory-of-Critical-Phenomena Preface_2018_A-Primer-to-the-Theory-of-Critical-Phenomena Preface TOC Chapter-1---Introduction--Classical-Phase_2018_A-Primer-to-the-Theory-of-Cri 1 Introduction: Classical Phases and Critical Points 1.1 Physical Examples: Macroscopic Properties 1.2 Singularity of the Specific Heat 1.3 Remarks Chapter-2---The-Ising-Model-and-Its-Basic-Chara_2018_A-Primer-to-the-Theory- 2 The Ising Model and Its Basic Characteristics in the Mean Field Approximation 2.1 The Ising Model and Its Hamiltonian 2.2 The Ising Partition Function and Free Energy 2.3 Equilibrium Conditions and Thermodynamics 2.4 Elementary Consequences of the Model 2.5 Discussion: Critical Exponents 2.6 The Meaning of the Ising Model and Mean Field Theory 2.7 Problem 2.1: Numerical Checkout 2.8 Problem 2.2: A Physical Estimate of Tc-Ordering Energy vs. Entropy Appendix 2.A Primer in Statistical Physics - A Summary 2.A.1 Principles of Thermodynamics (Thermostatics) 2.A.2 Connection to Statistical Physics 2.A.3 Elementary Example: Statistical Distribution Function for Quantum Multiparticle System: Fermions and Bosons 2.A.4 Outlook 2.A.5 Final Note Appendix 2.B Expansion of Eq. (2.5) in Powers of M Chapter-3---General-Mean-Field-Ap_2018_A-Primer-to-the-Theory-of-Critical-Ph 3 General Mean Field Approach 3.1 The Heisenberg Model and the Mean Field Approximation 3.2 Critical Phenomena in the Mean Field Approximation: A Further Elaboration 3.3 Van der Waals Equation of State and Criticality* 3.4 Density Fluctuations and Compressibility 3.5 Mean Field Theory for Binary Mixtures* Appendix 3.A Useful Relations Related to the van der Waals Equation Appendix 3.B Definition of Correlation Functions Appendix 3.C Gibbs-Duhem Rules Appendix 3.D Formula for the Entropy Part in Eq. (3.61) Chapter-4---The-Landau-Theory-of-Phase-Transitions-_2018_A-Primer-to-the-The 4 The Landau Theory of Phase Transitions: General Concept and Its Microscopic Relation to Mean Field Theory 4.1 Concept of the Order Parameter 4.2 Spin Magnetism: Microscopic Derivation of the Landau Theory with Spatial Fluctuations of the Order Parameter* 4.2.1 The Concept of Exchange Fields and Ferromagnetism 4.2.2 The Appearance of the Gradient Term 4.2.3 Formal Equivalence of the Landau Expansion and the Mean-field Approach 4.2.4 Landau Theory of the Ferromagnetic Transition Supplement: Zero-field Susceptibility in the Ferromagnetic Phase 4.2.5 Mean-field Theory for Two-sublattice Antiferromagnets 4.3 Rotationally Invariant Form of the Landau Functional 4.4 Outlook: Meaning of Mean Field Theory Problem 4.1: A Simple Physical Example of a Continuous Phase Transition: Order-disorder Transformation Problem 4.2: The Entropy and a Continuous Phase Transition for AB Alloys 4.5 Historical Note: Order of the Phase Transition Chapter-5---More-General-Considerations-Concerning_2018_A-Primer-to-the-Theo 5 More General Considerations Concerning Mean Field Theory: The Stratonovich-Hubbard Transformation* 5.1 General Form of the Partition Function for the Ising Model 5.2 The Upper Critical Dimension 5.3 Appendix: Derivation of Eq. (5.4) Chapter-6---Generalities-Relating-to-the-St_2018_A-Primer-to-the-Theory-of-C 6 Generalities Relating to the Study of Critical Phenomena 6.1 Introduction 6.2 Scaling Procedures: Kadanoff Blocks 6.3 Operations in Reciprocal Space 6.4 Utility of Algebraic Power Laws 6.5 Homogeneous Functions 6.6 Fixed Points 6.7 Appendix: A Simple Example Chapter-7---Failure-of-Mean-Field-Theory-_2018_A-Primer-to-the-Theory-of-Cri 7 Failure of Mean Field Theory and Scaling Methods 7.1 Kadanoff Scaling 7.2 Properties of the Homogeneous Equation 7.3 Scaling Laws for Second Derivatives 7.4 Summary and Remarks 7.5 Appendix A: Scaling Law for the Chemical Potential Chapter-8---Kadanoff-Scaling_2018_A-Primer-to-the-Theory-of-Critical-Phenome 8 Kadanoff Scaling 8.1 Example Involving a Ring 8.2 The Case of Two Dimensions 8.3 Rescaling of the Triangular Lattice 8.4 Determination of Averaged Perturbation Potentials 8.5 Determination of the Fixed Point Chapter-9---The-Renormalization-Group_2018_A-Primer-to-the-Theory-of-Critica 9 The Renormalization Group Operations 9.1 Real Space Renormalization 9.2 Renormalization of the Hamiltonian 9.3 Tracking Parametric Changes During Coarse Graining 9.4 An Example 9.5 Group Properties of the Coarse Graining Operation 9.6 Scaling Fields and Properties 9.7 Classification of Variables and Related Fixed Points 9.8 Back to Homogeneity Relations 9.9 Consequences Appendix A Examples Chapter-10---Additional-Interrelations-Bet_2018_A-Primer-to-the-Theory-of-Cr 10 Additional Interrelations Between Critical Exponents 10.1 Magnetization and Correlation Effects 10.2 Effect of Applied Magnetic Fields 10.3 Magnetization Close to the Critical Temperature 10.4 Magnetic Susceptibility Near the Critical Temperature 10.5 Heat Capacity 10.6 Summary Chapter-11---Extension-of-the-Landau-Approach-to-Inh_2018_A-Primer-to-the-Th 11 Extension of the Landau Approach to Inhomogeneous Systems and the Physical Picture of Gaussian Fluctuations 11.1 Smoothing Operations 11.2 Use of Functional Integrals Interpretation of the Formalism 11.3 Rock-Bottom Approximation 11.4 Fourier Transforms and Scaling in Reciprocal Space 11.5 Extension: The Ginsburg-Landau Functional in a Rotationally Invariant Case 11.6 A Concrete Example: Gaussian Fluctuations and the Ginzburg Criterion Chapter-12---The-Ginzburg-Landau-Functional-for-a-_2018_A-Primer-to-the-Theo 12 The Ginzburg-Landau Functional for a Continuous System: Formal Approach to Gaussian Fluctuations 12.1 Gaussian Integrals 12.2 Partition Function for the Gaussian Model 12.3 Operations in k Space 12.4 Partition Function 12.5 Correlation Functions 12.6 Comment 12.7 Renormalization of the Hamiltonian 12.8 Establishing the Requirements of the Fixed Point 12.9 Conclusion Appendix 12.A Hubbard-Stratonovich Transformation Appendix 12.B Relation |∇xφ|2=-φ∇2φ Appendix 12.C Comment on Functional Integration Appendix 12.D Meaning of Functional (Variational) Derivatives Chapter-13---The-Ginzburg-Landau-Wilson-Formal_2018_A-Primer-to-the-Theory-o 13 The Ginzburg-Landau-Wilson Formalism: Beyond the Gaussian Approximation 13.1 Generalities 13.2 The RGT Execution 13.3 Rescaling Effects 13.4 Case d>4 13.5 Case d