Two Dimensional Ising Model 2nd edition by Barry McCoy, Tai Tsun Wu ISBN B0FS7XL442 978-0486783123

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ISBN 10: B0FS7XL442
ISBN 13: 978-0486783123
Author: Barry McCoy, Tai Tsun Wu 

“Of all the systems in statistical mechanics on which exact calculations have been performed,” declare the authors of this text, “the two-dimensional Ising model is not only the most thoroughly investigated; it is also the richest and most profound.” Originally published in 1973, this is the definitive survey of the Ising model, a mathematical model of ferromagnetism in statistical mechanics. This updated edition of the classic text features an extensive section on new developments. Geared toward advanced undergraduates and graduate students of physics, it is also suitable for physicists working in statistical mechanics and related fields.

Following a brief introductory chapter, the book explores statistical mechanics, the one-dimensional Ising model, dimer statistics, specific heat of Onsager’s lattice in the absence of a magnetic field, boundary specific heat and magnetization, and boundary spin-spin correlation functions. Subsequent chapters cover the correlation functions, Wiener-Hopf sum equations, spontaneous magnetization, behavior of the correlation functions, asymptotic expansion, and boundary hysteresis and spin probability functions. Two other chapters examine Ising models with random impurities in terms of specific heat and boundary effects. The book concludes with a new chapter examining developments in the field since 1973.

Two Dimensional Ising Model 2nd Table of contents:

I Introduction

1 Introduction

2 The Ising Model

3 Literature

II Statistical Mechanics

1 Introduction

2 Microcanonical Ensemble

3 Canonical Ensemble

4 Thermodynamics

5 The Thermodynamic Limit

6 Extensions of Statistical Mechanics

III The One-Dimensional Ising Model

1 Introduction

2 Partition Function

3 Spin-Spin Correlation Functions

IV Dimer Statistics

1 Introduction

2 The Pfaffian

3 Dimer Configurations on Lattices with Free Boundary Conditions

4 Dimer Configurations on Lattices with Cylindrical Boundary Conditions

5 Dimer Configurations on Lattices with Toroidal Boundary Conditions

6 Evaluation of the Pfaffians

7 The Thermodynamic Limit

V Specific Heat of Onsager’s Lattice in the Absence of a Magnetic Field

1 Introduction

2 The Partition Function for Onsager’s Lattice

3 The Specific Heat of Onsager’s Lattice

4 The Antiferromagnetic Seam

VI Boundary Specific Heat and Magnetization

1 Introduction

2 Formulation of the Problem

3 Partition Function

4 Boundary Free Energy and Specific Heat

5 Boundary Magnetization

VII Boundary Spin-Spin Correlation Functions

1 Introduction

2 Formulation of the Problem

3 The Inverse of A

4 General Considerations

5 T > TC, N|1 − T/Tc| » 1

6 T 0, N[1 − T/Tc] » 1

7 T < TC, E1 < 0

8 T Near Tc

VIII The Correlation Functions ⟨σ0,0σ0,N⟩ and ⟨σ0,0σN,N⟩

1 Introduction

2 ⟨σ0,0σ0,N⟩ in Terms of an N × N Toeplitz Determinant

3 ⟨σ0,0σN,N⟩ in Terms of an N × N Toeplitz Determinant

4 The Near-Neighbor Correlation Functions ⟨σ0,0σ0,1⟩ and ⟨σ0,0σ1,1⟩

IX Wiener-Hopf Sum Equations

1 Introduction

2 Mathematical Technicalities

3 Factorization

4 Ind C(ξ) = 0

5 Ind C(ξ) < 0

6 Ind C(ξ) > 0

X Spontaneous Magnetization

1 Introduction

2 Discovering Szegö’s Theorem

3 A Rigorous Proof of Szegö’s Theorem

4 Explicit Calculation of the Spontaneous Magnetization

XI Behavior of the Correlation Functions ⟨σ0,0σ0,N⟩ and ⟨σ0,0σN,N⟩ for Large N

1 Introduction

2 Spin Correlations Above the Critical Temperature

3 Spin Correlations Below the Critical Temperature

4 ⟨σ0,0σN,N⟩ at T = Tc

5 ⟨σ0,0σ0,N⟩ at T = Tc; Leading Term

6 ⟨σ0,0σ0,N⟩ at T = Tc; Higher-Order Terms (I)

7 ⟨σ0,0σ0,N⟩ at T = Tc; Higher-Order Terms (II)

XII Asymptotic Expansion of ⟨σ0,0σM,N⟩

1 Introduction

2 The Correlation ⟨σ0,0σM,N⟩

3 Spin Correlations Below the Critical Temperature

4 Spin Correlations Above the Critical Temperature

5 Discussion

XIII Boundary Hysteresis and Spin Probability Functions

1 Introduction

2 Boundary Hysteresis: A Crude Interpretation

3 Critical Isotherm

4 Boundary Hysteresis: A Refined Interpretation

5 Misfit Bond

XIV An Ising Model with Random Impurities: Specific Heat

1 Introduction

2 Formulation of the Problem

3 Integral equation for v(x)

4 Power-Law Distribution

5 Discussion

XV An Ising Model with Random Impurities: Boundary Effects

1 Introduction

2 Average Boundary Magnetization

3 Average Spin-Spin Correlation Functions

XVI Epilogue

XVII Developments Since 1973

1 Introduction

2 Scaling Theory

3 Expansion of Correlation Functions

4 Painlevé Equations

5 Nonlinear Partial Difference Equations

6 The Magnetic Susceptibility

7 The Ising Model for H ≠ 0

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