Computational Studies of
Pure and Dilute Spin Models

by Peter J. Meyer

Title Page

Abstract

Acknowledgements

Chapter 1:  Spin Models
1.1   The Role of Models in Physics
1.2   Computational Physics
1.3   Modelling Magnetic Material
1.4   Lattice Geometries
1.5   Ising and Potts Spin Models
1.6   Dynamics and the Principle of Detailed Balance
1.7   Single Spin Flip Dynamics Algorithms
1.8   Temperature
1.9   Cluster Flip Dynamics Algorithms
1.10 Time
1.11 Boundary Conditions
1.12 Finite-Size Effects
Chapter 2: Validation of the Simulation Program
2.1   The Simulation Program
2.2   Comparison of Results for Internal Energy and Specific Heat
         with Those from High Temperature Series Expansion
2.3   Comparison of Simulation Results from Different Dynamics Algorithms
Chapter 3: Critical Temperatures of Pure Ising Spin Models
3.1   The Binder Cumulant
3.2   Thermal Equilibrium
3.3   The Critical Temperatures of Three Pure 2d Lattices
3.4   The Critical Temperatures of Two Pure 3d Lattices
3.5   The Critical Temperature of the Pure 4d Hypercubic Lattice
Chapter 4: Critical Temperatures of Dilute Ising Spin Models.
4.1   Thermal Equilibrium in Dilute Systems
4.2   A Critical Temperature of the Ising Cubic Lattice with Site Dilution
4.3   Dependence of Critical Temperature on Site Dilution in the Ising Square Lattice
4.4   Dependence of Critical Temperature on Bond Dilution in the Ising Square Lattice
Chapter 5: Extraction of the Critical Exponent of the Magnetization.
5.1   Introduction
5.2   The Ising Model on the Square Lattice
5.3   The Ising Model on the Triangular Lattice
5.4   The Ising Model on the Honeycomb Lattice
5.5   The 3-state Potts Model on the Square Lattice
Chapter 6: Short-Time Critical Dynamics
6.1   Introduction
6.2   The Ising Model
6.3   The 3-state Potts Model
6.4   The 4-state Potts Model
Chapter 7: Summary of Results and Comparison with Published Results

Chapter 8: Extensions of this Work
8.1   Development of the Simulation Software
8.2   Simulated Annealing

Appendix 1: Calculation of the Metropolis and the Glauber Transition Probabilities for the Ising Model and for the q-state Potts Model

Appendix 2: The Random Number Generator

Appendix 3: Contents of the Disk

Appendix 4: The FIT Programs

Appendix 5: Data Plotted in Graphs

Appendix 6: Site and Bond Percolation Thresholds

Appendix 7: Determination of Correlation Lengths

Appendix 8: Relaxation Time and Singular Dynamic Scaling

References

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