Lectures on Phase Transitions and the Renormalization Group

Cover
Half Title
Title Page
Copyright Page
Dedication
Editor\'s Foreword
Table of Contents
Preface
Chapter 1: Introduction
	1.1 Scaling and Dimensional Analysis
	1.2 Power Laws in Statistical Physics
		1.2.1 Liquid Gas Critical Point
		1.2.2 Magnetic Critical Point
		1.2.3 Superfluid Transition in 4He
		1.2.4 Self-Avoiding Walk
		1.2.5 Dynamic Critical Phenomena
	1.3 Some Important Questions
	1.4 Historical Development
	Exercises
Chapter 2: How Phase Transitions Occur in Principle
	2.1 Review of Statistical Mechanics
	2.2 The Thermodynamic Limit
		2.2.1 Thermodynamic Limit in a Charged System
		2.2.2 Thermodynamic Limit for Power Law Interactions
	2.3 Phase Boundaries and Phase Transition
		2.3.1 Ambiguity in the Definition of Phase Boundary
		2.3.2 Types of Phase Transition
		2.3.3 Finite Size Effects and the Correlation Length
	2.4 The Role of Models
	2.5 The Ising Model
	2.6 Analytic Properties of the Ising Model
		2.6.1 Convex Functions
		2.6.2 Convexity and the Free Energy Density
	2.7 Symmetry Properties of the Ising Model
		2.7.1 Time Reversal Symmetry
		2.7.2 Sub-Lattice Symmetry
	2.8 Existence of Phase Transitions
		2.8.1 Zero Temperature Phase Diagram
		2.8.2 Phase Diagram at Non-Zero Temperature: d = 1
		2.8.3 Phase Diagram at Non-Zero Temperature: d = 2
		2.8.4 Impossibility of Phase Transitions
	2.9 Spontaneous Symmetry Breaking
		2.9.1 Probability Distribution
		2.9.2 Continuous Symmetry
	2.10 Ergodicity Breaking
		2.10.1 Illustrative Example
		2.10.2 Symmetry and Its Implications for Ergodicity Breaking
		2.10.3 Example of Replica Symmetry Breaking: Rubber
		2.10.4 Order Parameters and Overlaps in a Classical Spin Glass
		2.10.5 Replica Formalism for Constrained Systems
	2.11 Fluids
	2.12 Lattice Gases
		2.12.1 Lattice Gas Thermodynamics from the Ising Model
		2.12.2 Derivation of Lattice Gas Model from the Configurational Sum
	2.13 Equivalence in Statistical Mechanics
	2.14 Miscellaneous Remarks
		2.14.1 History of the Thermodynamic Limit
		2.14.2 Do Quantum Effects Matter?
	Exercises
Chapter 3: How Phase Transitions Occur in Practice
	3.1 Ad Hoc Solution Methods
		3.1.1 Free Boundary Conditions and H = 0
		3.1.2 Periodic Boundary Conditions and H = 0
		3.1.3 Recursion Method for H = 0
		3.1.4 Effect of Boundary Conditions
	3.2 The Transfer Matrix
	3.3 Phase Transitions
	3.4 Thermodynamic Properties
	3.5 Spatial Correlations
		3.5.1 Zero Field: Ad Hoc Method
		3.5.2 Existence of Long Range Order
		3.5.3 Transfer Matrix Method
	3.6 Low Temperature Expansion
		3.6.1 d > 1
		3.6.2 d = 1
	3.7 Mean Field Theory
		3.7.1 Weiss’ Mean Field Theory
		3.7.2 Spatial Correlations
		3.7.3 How Good Is Mean Field Theory?
	Exercises
Chapter 4: Critical Phenomena in Fluids
	4.1 Thermodynamics
		4.1.1 Thermodynamic Potentials
		4.1.2 Phase Diagram
		4.1.3 Landau’s Symmetry Principle
	4.2 Two-Phase Coexistence
		4.2.1 Fluid at Constant Pressure
		4.2.2 Fluid at Constant Temperature
		4.2.3 Maxwell’s Equal Area Rule
	4.3 Vicinity of the Critical Point
	4.4 Van der Waals’ Equation
		4.4.1 Determination of the Critical Point
		4.4.2 Law of Corresponding States
		4.4.3 Critical Behaviour
	4.5 Spatial Correlations
		4.5.1 Number Fluctuations and Compressibility
		4.5.2 Number Fluctuations and Correlations
		4.5.3 Critical Opalescence
		4.6 Measurement of Critical Exponents
		4.6.1 Definition of Critical Exponents
		4.6.2 Determination of Critical Exponents
	Exercises
Chapter 5: Landau Theory
	5.1 Order Parameters
		5.1.1 Heisenberg Model
		5.1.2 XY Model
		5.1.3 3He
	5.2 Common Features of Mean Field Theories
	5.3 Phenomenological Landau Theory
		5.3.1 Assumptions
		5.3.2 Construction of ℒ
	5.4 Continuous Phase Transitions
		5.4.1 Critical Exponents in Landau Theory
	5.5 First Order Transitions
	5.6 Inhomogeneous Systems
		5.6.1 Coarse Graining
		5.6.2 Interpretation of the Landau Free Energy
	5.7 Correlation Functions
		5.7.1 The Continuum Limit
		5.7.2 Functional Integrals in Real and Fourier Space
		5.7.3 Functional Differentiation
		5.7.4 Response Functions
		5.7.5 Calculation of Two-Point Correlation Function
		5.7.6 The Coefficient γ
	Exercises
Chapter 6: Fluctuations and the Breakdown of Landau Theory
	6.1 Breakdown of Microscopic Landau Theory
		6.1.1 Fluctuations Away from the Critical Point
		6.1.2 Fluctuations Near the Critical Point
	6.2 Breakdown of Phenomenological Landau Theory
		6.2.1 Calculation of the Ginzburg Criterion
		6.2.2 Size o f the Critical Region
	6.3 The Gaussian Approximation
		6.3.1 One Degree of Freedom
		6.3.2 N Degrees of Freedom
		6.3.3 Infinite Number of Degrees of Freedom
		6.3.4 Two-Point Correlation Function Revisited
	6.4 Critical Exponents
	Exercises
Chapter 7: Anomalous Dimensions
	7.1 Dimensional Analysis of Landau Theory
	7.2 Dimensional Analysis and Critical Exponents
	7.3 Anomalous Dimensions and Asymptotics
	7.4 Renormalisation and Anomalous Dimensions
	Exercises
Chapter 8: Scaling in Static, Dynamic and Non-Equilibrium Phenomena
	8.1 The Static Scaling Hypothesis
		8.1.1 Time-Reversal Symmetry
		8.1.2 Behaviour as h → 0
		8.1.3 The Zero-held Susceptibility
		8.1.4 The Critical Isotherm and a Scaling Law
	8.2 Other Forms of the Scaling Hypothesis
		8.2.1 Scaling Hypothesis for the Free Energy
		8.2.2 Scaling Hypothesis for the Correlation Function
		8.2.3 Scaling and the Correlation Length
	8.3 Dynamic Critical Phenomena
		8.3.1 Small Time-Dependent Fluctuations
		8.3.2 The Relaxation Time
		8.3.3 Dynamic Scaling Hypothesis for Relaxation Times
		8.3.4 Dynamic Scaling Hypothesis for the Response Function
		8.3.5 Scaling of the Non-Unear Response
	8.4 Scaling in the Approach to Equilibrium
		8.4.1 Growth of a Fluctuating Surface
		8.4.2 Spinodal Decomposition in AUoys and Block Copolymers
	8.5 Summary
	Appendix 8 The Fokker-Planck Equation
Chapter 9: The Renormalisation Group
	9.1 Block Spins
		9.1.1 Thermodynamics
		9.1.2 Correlation Functions
		9.1.3 Discussion
	9.2 Basic Ideas of the Renormalisation Group
		9.2.1 Properties of Renormalisation Group Transformations
		9.2.2 The Origin of Singular Behaviour
	9.3 Fixed Points
		9.3.1 Physical Significance of Fixed Points
		9.3.2 Local Behaviour of RG Flows Near a Fixed Point
		9.3.3 Global Properties of RG Flows
	9.4 Origin of Scaling
		9.4.1 One Relevant Variable
		9.4.2 Diagonal RG Transformation for Two Relevant Variables
		9.4.3 Irrelevant Variables
		9.4.4 Non-diagonal RG Transformations
	9.5 RG in Differential Form
	9.6 RG for the Two Dimensional Ising Model
		9.6.1 Exact Calculation of Eigenvalues from Onsager’s Solution
		9.6.2 Formal Representation of the Coarse-Grained Hamiltonian
		9.6.3 Perturbation Theory for the RG Recursion Relation
		9.6.4 Fixed Points and Critical Exponents
		9.6.5 Effect of External Field
		9.6.6 Phase Diagram
		9.6.7 Remarks
	9.7 First Order Transitions and Non-Critical Properties
	9.8 RG for the Correlation Function
	9.9 Crossover Phenomena
		9.9.1 Small Fields
		9.9.2 Crossover Arising from Anisotropy
		9.9.3 Crossover and Disorder: the Harris Criterion
	9.10 Corrections to Scaling
	9.11 Finite Size Scaling
	Exercises
Chapter 10: Anomalous Dimensions Far From Equilibrium
	10.1 Introduction
	10.2 Similarity Solutions
		10.2.1 Long Time Behaviour of the Diffusion Equation
		10.2.2 Dimensional Analysis of the Diffusion Equation
		10.2.3 Intermediate Asymptotics of the First Kind
	10.3 Anomalous Dimensions in Similarity Solutions
		10.3.1 The Modified Porous Medium Equation
		10.3.2 Dimensional Analysis for Barenblatt’s Equation
		10.3.3 Similarity Solution with an Anomalous Dimension
		10.3.4 Intermediate Asymptotics of the Second Kind
	10.4 Renormalisation
		10.4.1 Renormalisation and its Physical Interpretation
		10.4.2 Heuristic Calculation of the Anomalous Dimension α
		10.4.3 Renormalisation and Dimensional Analysis
		10.4.4 Removal of Divergences and the RG
		10.4.5 Assumption of Renormalisability
		10.4.6 Renormalisation and Physical Theory
		10.4.7 Renormalisation of the Modified Porous Medium Equation
	10.5 Perturbation Theory for Barenblatt’s Equation
		10.5.1 Formal Solution
		10.5.2 Zeroth Order in ϵ
		10.5.3 First Order in ϵ
		10.5.4 Isolation of the Divergence
		10.5.5 Perturbative Renormalisation
		10.5.6 Renormalised Perturbation Expansion
		10.5.7 Origin of Divergence of Perturbation Theory
	10.6 Fixed Points
		10.6.1 Similarity Solutions as Fixed Points
		10.6.2 Universality in the Approach to Equilibrium
	10.7 Conclusion
	Appendix 10 Method of Characteristics
	Exercises
Chapter 11: Continuous Symmetry
	11.1 Correlation in the Ordered Phase
		11.1.1 The Susceptibility Tensor
		11.1.2 Excitations for T < Te: Goldstone’s Theorem
		11.1.3 Emergence of Order Parameter Rigidity
		11.1.4 Scaling of the Stiffness
		11.1.5 Lower Critical Dimension
	11.2 Kosterlitz-Thouless Transition
		11.2.1 Phase Fluctuations
		11.2.2 Phase Correlations
		11.2.3 Vortex Unbinding
		11.2.4 Universal Jump in the Stiffness
Chapter 12: Critical Phenomen a Near Four Dimensions
	12.1 Basic Idea of the Epsilon Expansion
	12.2 RG for the Gaussian Mode
		12.2.1 Integrating Out the Short Wavelength Degrees of Freedom
		12.2.2 Rescaling of Fields and Momenta
		12.2.3 Analysis of Recursion Relation
		12.2.4 Critical Exponents
		12.2.5 A Dangerous Irrelevant Variable in Landau Theory
	12.3 RG Beyond the Gaussian Approximation
		12.3.1 Setting Up Perturbation Theory
		12.3.2 Calculation of 〈V〉0: Strategy
		12.3.3 Correlation Functions of σ̂ℓ(k) Wick’s Theorem
		12.3.4 Evaluation of 〈V〉0
	12.4 Feynman Diagrams
		12.4.1 Feynman Diagrams to O〈V〉
		12.4.2 Feynman Diagrams for 〈V〉0 — 〈V〉20
		12.4.3 Elimination of Unnecessary Diagrams
	12.5 The RG Recursion Relations
		12.5.1 Feynman Diagrams for Small ϵ = 4 – d
		12.5.2 Recursion Relations to O(ϵ)
		12.5.3 Fixed Points to O(ϵ)
		12.5.4 RG Flows and Exponents
	12.6 Conclusion
	Appendix 12 The Linked Cluster Theorem
	Exercises
Index