Statistical Mechanics of Phases and Phase Transitions (eBook)

Name: Statistical Mechanics of Phases and Phase Transitions (eBook)
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Preface
A Notational Note
Acknowledgements
1 Overview
- 1.1 Phases
  - 1.1.1 Phase Diagrams
  - 1.1.2 Phase Transitions
  - 1.1.3 The Critical Point
  - 1.1.4 Continuous and Discontinuous Transitions
  - 1.1.5 Distinction between Phases
  - 1.2 Symmetries
    - 1.2.1 The Cubic-to-Tetragonal Transition
    - 1.2.2 Solids, Liquids, and Gases
    - 1.2.3 Ferromagnets
    - 1.3 Universality
      - 1.3.1 Critical Exponents
      - 1.3.2 Universality Classes
      - 1.4 Problems
      - 1.5 Worksheets for Chapter 1
        W1.1 Critical Exponents
        2 Background: Fundamentals of Statistical Mechanics
        2.1 Review of Statistical Mechanics
        2.1.1 The Link between Thermodynamics and Statistical Mechanics
        2.1.2 The Thermodynamic Limit
        2.1.3 Derivatives of the Free Energy
        2.1.4 Variance and Fluctuations
        2.1.5 Covariance and Correlation Functions
        2.1.6 Correlation Functions and Phases of Matter
        2.2 *Other Fundamental Issues Concerning Statistical Mechanics
        2.2.1 Ergodicity
        2.3 Symmetry
        2.3.1 Meaning of "Symmetry"
        2.3.2 Symmetries in Statistical Mechanics
        2.4 Problems
        2.5 Worksheets for Chapter 2
        W2.1 Calculating Thermal Averages
        W2.2 Identifying Symmetries
        3 The Ising Model
        3.1 De¿nition
        3.1.1 What We Would Like to Know
        3.2 Qualitative Properties
        3.2.1 The Droplet Argument in d= 2
        3.2.2 Low-TSeries Expansion in Arbitrary d
        3.3 Solution in Zero Dimensions
        3.3.1 J= 0 Limit in dDimensions
        3.4 Solution in One Dimension
        3.4.1 h= 0 Solution-The Combinatorics Approach
        3.4.2 The General Solution Using the Transfer Matrix Approach
        3.5 Solution in Two Dimensions
        3.6 Variations on the Ising Model
        3.7 *Low TExpansion, High TExpansion, and Duality
        3.7.1 Low-Temperature Expansion to All Orders
        3.7.2 High-TSeries Expansion in Arbitrary d
        3.7.3 Duality
        3.8 Problems
        3.9 Worksheets for Chapter 3
        W3.1 The Ising Hamiltonian-A Primer
        W3.2 More on the 4-Site Problem
        W3.3 The Droplet Argument in 2D
        W3.4 Low-Temperature Expansion in 2D
        W3.5 Ising Model in 0D
        W3.6 Ising Model in 1D-The Combinatorics Approach
        W3.7 Ising Model in 1D-The Transfer Matrix Approach
        4 Mean-Field Theory
        4.1 Intuitive Introduction
        4.2 The Variational Principle
        4.2.1 Usefulness of the Variational Approach
        4.2.2 An Illustration of the Principle
        4.3 Mean-Field Theory of the Ising Model
        4.3.1 The Mean-Field Phase Diagram
        4.4 Variational Free Energy as Function of mj
        4.4.1 Expansion for Small m
        4.4.2 Critical Exponents
        4.4.3 *Metastability and Two-Phase Coexistence
        4.5 Validity of Mean-Field Theory
        4.6 Problems
        4.7 Worksheets for Chapter 4
        W4.1 Weiss Mean-Field Theory
        W4.2 Variational Principle
        W4.3 *Variational Mean-Field Theory-General Features
        W4.4 Mean-Field Solution of the Ising Model
        W4.5 Low Temperature Analysis of Mean-Field Equations
        W4.6 The Variational Free Energy Fvar as a Function of the Order Parameter
        W4.7 The Mean-Field Critical Exponents ß,¿,d
        5 Broken Symmetries
        5.1 Spontaneously Broken Symmetry and Order Parameters
        5.1.1 Why the Thermodynamic Limit Is Necessary
        5.1.2 Phase Transitions in the Real World
        5.2 Examples of Broken Symmetries
        5.2.1 Order-Disorder Transitions
        5.2.2 Structural Transitions
        5.2.3 Crystalline Solids
        5.2.4 Liquid Crystals
        5.2.5 Ferromagnetism
        5.2.6 Super¿uids
        5.2.7 Superconductors
        5.3 Order Parameters and Symmetry
        5.3.1 Thermodynamic Role of the Order Parameter
        5.4 Some Paradigmatic Models
        5.4.1 Models with Discrete Symmetries
        5.4.2 Models with Continuous Symmetries
        5.5 Problems
        5.6 Worksheets for Chapter 5
        W5.1 Liquid Crystals and Broken Symmetries
        W5.2 Interpolating between the XY and Clock Models
        W5.3 Determining the Symmetry and Order Parameter
        6 The Landau Free Energy
        6.1 Once More, the Ising Model
        6.2 Other Symmetries
        6.2.1 The XY Model: O(2) Symmetry
        6.2.2 q-State Clock Models
        6.3 Multicritical Points
        6.3.1 Tricritical Points
        6.3.2 Bicritical and Tetracritical Points
        6.4 Problems
        6.5 Worksheets for Chapter 6
        W6.1 Deducing the Landau Free Energy from Symmetry Considerations
        W6.2 Understanding the Discontinuous Transition with a ¿3 Term
        7 Ginzburg-Landau Theory
        7.1 The Thermodynamic Correlation Length
        7.2 Spatial Correlations in the Ising Model
        7.2.1 Position-Dependent Correlation Functions
        7.2.2 Calculating the Nonlocal Susceptibility ¿ij
        7.3 The E¿ective Field Theory
        7.3.1 Taking the Continuum Limit
        7.3.2 The Equilibrium Order Parameter Field
        7.3.3 Correlations in the Ordered Phase
        7.4 Beyond the Ising Model
        7.5 Order Parameter Textures
        7.5.1 Domain Walls in the Ising Ferromagnet
        7.5.2 Textures in the XY Model
        7.5.3 Vortices in the XY Model
        7.6 Equilibrium Fluctuations
        7.7 Problems
        7.8 Worksheets for Chapter 7
        W7.1 The Meaning of the Correlation Length
        W7.2 Ising Model with Spatial Variation
        W7.3 Estimating a Correlation Length from the Lattice Correlation Function
        W7.4 Deriving the Ginzburg-Landau Functional from Symmetry Arguments
        W7.5 Solving for the Continuum Correlation Length
        W7.6 Susceptibility of the XY Model for T < Tc
        W7.7 A Domain Wall in an Ising System
        8 *The Self-Consistent Gaussian Approximation
        8.1 The Gaussian Model
        8.1.1 The Nature of the Solution
        8.1.2 Higher Order Correlations and Wick's Theorem
        8.2 Using the Gaussian Model in a Variational Approximation
        > Tc
        8.3 ∗Exact Results for O(N) Model as N→oo
        8.3.1 The Ordered Phase and the Mermin-Wagner Theorem
        8.4 Worksheets for Chapter 8
        W8.1 Gaussian Integrals and Wick's Theorem (Isserlis's Theorem)
        W8.2 Mean-Field Approximations to the 0D ¿4 Theory
        9 *Quenched Disorder
        9.1 Systems with Quenched Disorder
        9.2 Con¿guration Averages
        9.2.1 Example: Array of Harmonic Oscillators
        9.3 Gaussian Model in a Random Field
        9.4 The Imry-Ma Argument
        9.5 Larkin-Ovchinikov-Lee-Rice Argument
        9.6 Other E¿ects of Disorder
        10 *Renormalization Group Perspective
        10.1 Scaling Theory
        10.1.1 Relevant and Irrelevant Perturbations
        10.2 Renormalization Group-Overview
        10.2.1 The Nature of the RG Transformation
        10.3 RG for the Central Limit Theorem
        10.4 The RG in Slightly Less than 4D
        10.5 Beyond Broken Symmetry: Superconductors, Topological Order, Gauge Theories
        10.6 Problems
        10.7 Worksheets for Chapter 10
        W10.1 Renormalization Group of the 1D Ising Chain
        W10.2 Perturbative Renormalization Group of the ¿4 Theory in dDimensions
        A Appendices
        A.1 The Qubit and Pauli Matrices
        A.2 Low-Temperature Expansion of the Ising Model
        A.3 The Fourier Series and Transforms
        A.4 The Calculus of Variations
        A.5 The Continuum Limit
        A.6 Fourier Transform of a Lorentzian: The Ornstein-Zernike Law
        A.6.1d= 1
        A.6.2 d= 3
        A.6.3 Arbitrary d
        A.6.4 At Criticality: Orenstein-Zernike as ¿→ 0.
        A.7 Review of Probability
        A.8 Deriving the Variational Principle for Noncommuting Hamiltonians
        A.9 Properties of Gaussian Distributions and Wick's Theorem
        A.10 Rudiments of Group Theory
        A.10.1 Symmetries as a Group: An Example in a Triangle
        A.11 Validity of Mean-Field Theory in an Ising Ferromagnet with Long-Range Interactions
        A.12 More about the Broken Symmetry in Super¿uids
        B Worksheet Solutions
        B.1 Chapter 1 Worksheet Solutions
        B.1.1 Critical Exponents
        B.2 Chapter 2 Worksheet Solutions
        B.2.1 Calculating Thermal Averages
        B.2.2 Identifying Symmetries
        B.3 Chapter 3 Worksheet Solutions
        B.3.1 The Ising Hamiltonian-A Primer
        B.3.2 More on the 4-Site Problem
        B.3.3 The Droplet Argument in 2D
        B.3.4 Low-Temperature Expansion in 2D
        B.3.5 Ising Model in 0D
        B.3.6 Ising Model in 1D-The Combinatorics Approach
        B.3.7 Ising Model in 1D-The Transfer Matrix Approach
        B.4 Chapter 4 Worksheet Solutions
        B.4.1 Weiss Mean-Field Theory
        B.4.2 Variational Principle
        B.4.3 *Variational Mean-Field Theory-General Features
        B.4.4 Mean-Field Solution of the Ising Model
        B.4.5 Low Temperature Analysis of Mean-Field Equations
        B.4.6 The Variational Free Energy Fvar as a Function of the Order Parameter
        B.4.7 The Mean-Field Critical Exponents ß,¿,d
        B.5 Chapter 5 Worksheet Solutions
        B.5.1 Liquid Crystals and Broken Symmetries
        B.5.2 Interpolating between the XY and Clock Models
        B.5.3 Determining the Symmetry and Order Parameter
        B.6 Chapter 6 Worksheet Solutions
        B.6.1 Deducing the Landau Free Energy from Symmetry Considerations
        B.6.2 Understanding the Discontinuous Transition with a ¿3 Term
        B.7 Chapter 7 Worksheet Solutions
        B.7.1 The Meaning of the Correlation Length
        B.7.2 Ising Model with Spatial Variation
        B.7.3 Estimating a Correlation Length from the Lattice Correlation Function
        B.7.4 Deriving the Ginzburg-Landau Functional from Symmetry Arguments
        B.7.5 Solving for the Continuum Correlation Length
        B.7.6 Susceptibility of the XY Model for T < Tc
        B.7.7 A Domain Wall in an Ising System
        B.8 Chapter 8 Worksheet Solutions
        B.8.1 Gaussian Integrals and Wick's Theorem (Isserlis's Theorem)
        B.8.2 Mean-Field Approximations to the 0D ¿4 Theory
        B.10 Chapter 10 Worksheet Solutions
        B.10.1 Renormalization Group of the 1D Ising Chain
        B.10.2 Perturbative Renormalization Group of the ¿4 Theory in dDimensions
        Bibliography
        Index

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Preface
A Notational Note
Acknowledgements
1 Overview
- 1.1 Phases
  - 1.1.1 Phase Diagrams
  - 1.1.2 Phase Transitions
  - 1.1.3 The Critical Point
  - 1.1.4 Continuous and Discontinuous Transitions
  - 1.1.5 Distinction between Phases
  - 1.2 Symmetries
    - 1.2.1 The Cubic-to-Tetragonal Transition
    - 1.2.2 Solids, Liquids, and Gases
    - 1.2.3 Ferromagnets
    - 1.3 Universality
      - 1.3.1 Critical Exponents
      - 1.3.2 Universality Classes
      - 1.4 Problems
      - 1.5 Worksheets for Chapter 1
        W1.1 Critical Exponents
        2 Background: Fundamentals of Statistical Mechanics
        2.1 Review of Statistical Mechanics
        2.1.1 The Link between Thermodynamics and Statistical Mechanics
        2.1.2 The Thermodynamic Limit
        2.1.3 Derivatives of the Free Energy
        2.1.4 Variance and Fluctuations
        2.1.5 Covariance and Correlation Functions
        2.1.6 Correlation Functions and Phases of Matter
        2.2 *Other Fundamental Issues Concerning Statistical Mechanics
        2.2.1 Ergodicity
        2.3 Symmetry
        2.3.1 Meaning of "Symmetry"
        2.3.2 Symmetries in Statistical Mechanics
        2.4 Problems
        2.5 Worksheets for Chapter 2
        W2.1 Calculating Thermal Averages
        W2.2 Identifying Symmetries
        3 The Ising Model
        3.1 De¿nition
        3.1.1 What We Would Like to Know
        3.2 Qualitative Properties
        3.2.1 The Droplet Argument in d= 2
        3.2.2 Low-TSeries Expansion in Arbitrary d
        3.3 Solution in Zero Dimensions
        3.3.1 J= 0 Limit in dDimensions
        3.4 Solution in One Dimension
        3.4.1 h= 0 Solution-The Combinatorics Approach
        3.4.2 The General Solution Using the Transfer Matrix Approach
        3.5 Solution in Two Dimensions
        3.6 Variations on the Ising Model
        3.7 *Low TExpansion, High TExpansion, and Duality
        3.7.1 Low-Temperature Expansion to All Orders
        3.7.2 High-TSeries Expansion in Arbitrary d
        3.7.3 Duality
        3.8 Problems
        3.9 Worksheets for Chapter 3
        W3.1 The Ising Hamiltonian-A Primer
        W3.2 More on the 4-Site Problem
        W3.3 The Droplet Argument in 2D
        W3.4 Low-Temperature Expansion in 2D
        W3.5 Ising Model in 0D
        W3.6 Ising Model in 1D-The Combinatorics Approach
        W3.7 Ising Model in 1D-The Transfer Matrix Approach
        4 Mean-Field Theory
        4.1 Intuitive Introduction
        4.2 The Variational Principle
        4.2.1 Usefulness of the Variational Approach
        4.2.2 An Illustration of the Principle
        4.3 Mean-Field Theory of the Ising Model
        4.3.1 The Mean-Field Phase Diagram
        4.4 Variational Free Energy as Function of mj
        4.4.1 Expansion for Small m
        4.4.2 Critical Exponents
        4.4.3 *Metastability and Two-Phase Coexistence
        4.5 Validity of Mean-Field Theory
        4.6 Problems
        4.7 Worksheets for Chapter 4
        W4.1 Weiss Mean-Field Theory
        W4.2 Variational Principle
        W4.3 *Variational Mean-Field Theory-General Features
        W4.4 Mean-Field Solution of the Ising Model
        W4.5 Low Temperature Analysis of Mean-Field Equations
        W4.6 The Variational Free Energy Fvar as a Function of the Order Parameter
        W4.7 The Mean-Field Critical Exponents ß,¿,d
        5 Broken Symmetries
        5.1 Spontaneously Broken Symmetry and Order Parameters
        5.1.1 Why the Thermodynamic Limit Is Necessary
        5.1.2 Phase Transitions in the Real World
        5.2 Examples of Broken Symmetries
        5.2.1 Order-Disorder Transitions
        5.2.2 Structural Transitions
        5.2.3 Crystalline Solids
        5.2.4 Liquid Crystals
        5.2.5 Ferromagnetism
        5.2.6 Super¿uids
        5.2.7 Superconductors
        5.3 Order Parameters and Symmetry
        5.3.1 Thermodynamic Role of the Order Parameter
        5.4 Some Paradigmatic Models
        5.4.1 Models with Discrete Symmetries
        5.4.2 Models with Continuous Symmetries
        5.5 Problems
        5.6 Worksheets for Chapter 5
        W5.1 Liquid Crystals and Broken Symmetries
        W5.2 Interpolating between the XY and Clock Models
        W5.3 Determining the Symmetry and Order Parameter
        6 The Landau Free Energy
        6.1 Once More, the Ising Model
        6.2 Other Symmetries
        6.2.1 The XY Model: O(2) Symmetry
        6.2.2 q-State Clock Models
        6.3 Multicritical Points
        6.3.1 Tricritical Points
        6.3.2 Bicritical and Tetracritical Points
        6.4 Problems
        6.5 Worksheets for Chapter 6
        W6.1 Deducing the Landau Free Energy from Symmetry Considerations
        W6.2 Understanding the Discontinuous Transition with a ¿3 Term
        7 Ginzburg-Landau Theory
        7.1 The Thermodynamic Correlation Length
        7.2 Spatial Correlations in the Ising Model
        7.2.1 Position-Dependent Correlation Functions
        7.2.2 Calculating the Nonlocal Susceptibility ¿ij
        7.3 The E¿ective Field Theory
        7.3.1 Taking the Continuum Limit
        7.3.2 The Equilibrium Order Parameter Field
        7.3.3 Correlations in the Ordered Phase
        7.4 Beyond the Ising Model
        7.5 Order Parameter Textures
        7.5.1 Domain Walls in the Ising Ferromagnet
        7.5.2 Textures in the XY Model
        7.5.3 Vortices in the XY Model
        7.6 Equilibrium Fluctuations
        7.7 Problems
        7.8 Worksheets for Chapter 7
        W7.1 The Meaning of the Correlation Length
        W7.2 Ising Model with Spatial Variation
        W7.3 Estimating a Correlation Length from the Lattice Correlation Function
        W7.4 Deriving the Ginzburg-Landau Functional from Symmetry Arguments
        W7.5 Solving for the Continuum Correlation Length
        W7.6 Susceptibility of the XY Model for T < Tc
        W7.7 A Domain Wall in an Ising System
        8 *The Self-Consistent Gaussian Approximation
        8.1 The Gaussian Model
        8.1.1 The Nature of the Solution
        8.1.2 Higher Order Correlations and Wick's Theorem
        8.2 Using the Gaussian Model in a Variational Approximation
        > Tc
        8.3 ∗Exact Results for O(N) Model as N→oo
        8.3.1 The Ordered Phase and the Mermin-Wagner Theorem
        8.4 Worksheets for Chapter 8
        W8.1 Gaussian Integrals and Wick's Theorem (Isserlis's Theorem)
        W8.2 Mean-Field Approximations to the 0D ¿4 Theory
        9 *Quenched Disorder
        9.1 Systems with Quenched Disorder
        9.2 Con¿guration Averages
        9.2.1 Example: Array of Harmonic Oscillators
        9.3 Gaussian Model in a Random Field
        9.4 The Imry-Ma Argument
        9.5 Larkin-Ovchinikov-Lee-Rice Argument
        9.6 Other E¿ects of Disorder
        10 *Renormalization Group Perspective
        10.1 Scaling Theory
        10.1.1 Relevant and Irrelevant Perturbations
        10.2 Renormalization Group-Overview
        10.2.1 The Nature of the RG Transformation
        10.3 RG for the Central Limit Theorem
        10.4 The RG in Slightly Less than 4D
        10.5 Beyond Broken Symmetry: Superconductors, Topological Order, Gauge Theories
        10.6 Problems
        10.7 Worksheets for Chapter 10
        W10.1 Renormalization Group of the 1D Ising Chain
        W10.2 Perturbative Renormalization Group of the ¿4 Theory in dDimensions
        A Appendices
        A.1 The Qubit and Pauli Matrices
        A.2 Low-Temperature Expansion of the Ising Model
        A.3 The Fourier Series and Transforms
        A.4 The Calculus of Variations
        A.5 The Continuum Limit
        A.6 Fourier Transform of a Lorentzian: The Ornstein-Zernike Law
        A.6.1d= 1
        A.6.2 d= 3
        A.6.3 Arbitrary d
        A.6.4 At Criticality: Orenstein-Zernike as ¿→ 0.
        A.7 Review of Probability
        A.8 Deriving the Variational Principle for Noncommuting Hamiltonians
        A.9 Properties of Gaussian Distributions and Wick's Theorem
        A.10 Rudiments of Group Theory
        A.10.1 Symmetries as a Group: An Example in a Triangle
        A.11 Validity of Mean-Field Theory in an Ising Ferromagnet with Long-Range Interactions
        A.12 More about the Broken Symmetry in Super¿uids
        B Worksheet Solutions
        B.1 Chapter 1 Worksheet Solutions
        B.1.1 Critical Exponents
        B.2 Chapter 2 Worksheet Solutions
        B.2.1 Calculating Thermal Averages
        B.2.2 Identifying Symmetries
        B.3 Chapter 3 Worksheet Solutions
        B.3.1 The Ising Hamiltonian-A Primer
        B.3.2 More on the 4-Site Problem
        B.3.3 The Droplet Argument in 2D
        B.3.4 Low-Temperature Expansion in 2D
        B.3.5 Ising Model in 0D
        B.3.6 Ising Model in 1D-The Combinatorics Approach
        B.3.7 Ising Model in 1D-The Transfer Matrix Approach
        B.4 Chapter 4 Worksheet Solutions
        B.4.1 Weiss Mean-Field Theory
        B.4.2 Variational Principle
        B.4.3 *Variational Mean-Field Theory-General Features
        B.4.4 Mean-Field Solution of the Ising Model
        B.4.5 Low Temperature Analysis of Mean-Field Equations
        B.4.6 The Variational Free Energy Fvar as a Function of the Order Parameter
        B.4.7 The Mean-Field Critical Exponents ß,¿,d
        B.5 Chapter 5 Worksheet Solutions
        B.5.1 Liquid Crystals and Broken Symmetries
        B.5.2 Interpolating between the XY and Clock Models
        B.5.3 Determining the Symmetry and Order Parameter
        B.6 Chapter 6 Worksheet Solutions
        B.6.1 Deducing the Landau Free Energy from Symmetry Considerations
        B.6.2 Understanding the Discontinuous Transition with a ¿3 Term
        B.7 Chapter 7 Worksheet Solutions
        B.7.1 The Meaning of the Correlation Length
        B.7.2 Ising Model with Spatial Variation
        B.7.3 Estimating a Correlation Length from the Lattice Correlation Function
        B.7.4 Deriving the Ginzburg-Landau Functional from Symmetry Arguments
        B.7.5 Solving for the Continuum Correlation Length
        B.7.6 Susceptibility of the XY Model for T < Tc
        B.7.7 A Domain Wall in an Ising System
        B.8 Chapter 8 Worksheet Solutions
        B.8.1 Gaussian Integrals and Wick's Theorem (Isserlis's Theorem)
        B.8.2 Mean-Field Approximations to the 0D ¿4 Theory
        B.10 Chapter 10 Worksheet Solutions
        B.10.1 Renormalization Group of the 1D Ising Chain
        B.10.2 Perturbative Renormalization Group of the ¿4 Theory in dDimensions
        Bibliography
        Index

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Verlag	Princeton University Press
Einband	PDF
Erscheinungsjahr	2024
Seitenangabe	472 S.
Ausgabekennzeichen	Englisch
Abbildungen	49 color illus.
Plattform	PDF
Autor	Kivelson, Steven A. / Jiang, Jack Mingde / Chang, Jeffrey

Über den Autor Steven A. Kivelson

Steven A. Kivelson is the Prabhu Goel Family Professor of Physics at Stanford University. Jack Mingde Jiang received a PhD in applied physics from Stanford. Jeffrey Chang is a PhD student in the Department of Physics at Harvard University.