The Monte Carlo Method in Condensed Matter Physics

1. Introduction.- 1.1 General Remarks.- 1.2 Progress in the Understanding of Finite Size Effects at Phase Transitions.- 1.2.1 Asymmetric First-Order Phase Transition.- 1.2.2 Coexisting Phases.- 1.2.3 Critical Phenomena Studies in the Microcanonical Ensemble.- 1.2.4 Anisotropy Effects in Finite Size Scaling.- 1.3 Statistical Errors.- 1.4 Final Remarks.- References.- 2. Vectorisation of Monte Carlo Programs for Lattice Models Using Supercomputers.- 2.1 Introduction.- 2.2 Technical Details.- 2.2.1 Basic Principles.- 2.2.2 Some "Dos" and "Don'ts" of Vectorisation.- 2.3 Simple Vectorisation Algorithms.- 2.4 Vectorised Multispin Coding Algorithms.- 2.5 Vectorised Multilattice Coding Algorithms.- 2.6 Vectorised Microcanonical Algorithms.- 2.7 Some Recent Results from Vectorised Algorithms.- 2.7.1 Ising Model Critical Behaviour.- 2.7.2 First-Order Transitions in Potts Models.- 2.7.3 Dynamic Critical Behaviour.- 2.7.4 Surface and Interface Phase Transitions.- 2.7.5 Bulk Critical Behaviour in Classical Spin Systems.- 2.7.6 Quantum Spin Systems.- 2.7.7 Spin Exchange and Diffusion.- 2.7.8 Impurity Systems.- 2.7.9 Other Studies.- 2.8 Conclusion.- References.- 3. Parallel Algorithms for Statistical Physics Problems.- 3.1 Paradigms of Parallel Computing.- 3.1.1 Physics-Based Description.- (a) Event Parallelism.- (b) Geometric Parallelism.- (c) Algorithmic Parallelism.- 3.1.2 Machine-Based Description.- (a) SIMD Architecture.- (b) MIMD Architecture.- (c) The Connectivity.- (d) Measurements of Machine Performance.- 3.2 Applications on Fine-Grained SIMD Machines.- 3.2.1 Spin Systems.- 3.2.2 Molecular Dynamics.- 3.3 Applications on Coarse-Grained MIMD Machines.- 3.3.1 Molecular Dynamics.- 3.3.2 Cluster Algorithms for the Ising Model.- 3.3.3 Data Parallel Algorithms.- (a) Long-Range Interactions.- (b) Polymers.- 3.4 Prospects.- References.- 4. New Monte Carlo Methods for Improved Efficiency of Computer Simulations in Statistical Mechanics.- 4.1 Overview.- 4.2 Acceleration Algorithms.- 4.2.1 Critical Slowing Down and Standard Monte Carlo Method.- 4.2.2 Fortuin-Kasteleyn Transformation.- 4.2.3 Swendsen-Wang Algorithm.- 4.2.4 Further Developments.- 4.2.5 Replica Monte Carlo Method.- 4.2.6 Multigrid Monte Carlo Method.- 4.3 Histogram Methods.- 4.3.1 The Single-Histogram Method.- 4.3.2 The Multiple-Histogram Method.- 4.3.3 History and Applications.- 4.4 Summary.- References.- 5. Simulation of Random Growth Processes.- 5.1 Irreversible Growth of Clusters.- 5.1.1 A Simple Example of Cluster Growth: The Eden Model.- 5.1.2 Laplacian Growth.- (a) Moving Boundary Condition Problems.- (b) Numerical Simulation of Dielectric Breakdown and DLA.- (c) Fracture.- 5.2 Reversible Probabilistic Growth.- 5.2.1 Cellular Automata.- 5.2.2 Damage Spreading in the Monte Carlo Method.- 5.2.3 Numerical Results for the Ising Model.- 5.2.4 Heat Bath Versus Glauber Dynamics in the Ising Model.- 5.2.5 Relationship Between Damage and Thermodynamic Properties.- 5.2.6 Damage Clusters.- 5.2.7 Damage in Spin Glasses.- 5.2.8 More About Damage Spreading.- 5.3 Conclusion.- References.- 6. Recent Progress in the Simulation of Classical Fluids.- 6.1 Improvements of the Monte Carlo Method.- 6.1.1 Metropolis Algorithm.- 6.1.2 Monte Carlo Simulations and Statistical Ensembles.- (a) Canonical, Grand Canonical and Semi-grand Ensembles.- (b) Gibbs Ensemble.- (c) MC Algorithm for "Adhesive" Particles.- 6.1.3 Monte Carlo Computation of the Chemical Potential and the Free Energy.- (a) Chemical Potential.- (b) Free Energy.- 6.1.4 Algorithms for Coulombic and Dielectric Fluids.- 6.2 Pure Phases and Mixtures of Simple Fluids.- 6.2.1 Two-Dimensional Simple Fluids.- 6.2.2 Three-Dimensional Monatomic Fluids.- 6.2.3 Lennard-Jones Fluids and Similar Systems.- 6.2.4 Real Fluids.- 6.2.5 Mixtures of Simple Fluids.- (a) Hard Core Systems.- (b) LJ Mixtures.- (c) Polydisperse Fluids.- 6.3 Coulombic and Ionic Fluids.- 6.3.1 One-Component Plasma, Two-Component Plasma and Primitive Models of Electrolyte Solutions.- (a) OCP and TCP.- (b) Primitive Models.- 6.3.2 Realistic Ionic Systems.- 6.4 Simulations of Inhomogeneous Simple Fluids.- 6.4.1 Liquid-Vapour Interfaces.- 6.4.2 Fluid-Solid Interfaces.- 6.4.3 Interfaces of Charged Systems.- 6.4.4 Fluids in Narrow Pores.- 6.5 Molecular Liquids: Model Systems.- 6.5.1 Two-Dimensional Systems.- 6.5.2 Convex Molecules (Three-Dimensional).- (a) Virial Coefficients and the Equation of State.- (b) Pair Distribution Function.- (c) Phase Transitions.- 6.5.3 Site-Site Potentials.- 6.5.4 Chain Molecules.- 6.5.5 Dipolar Systems.- 6.5.6 Quadrupolar Systems.- 6.5.7 Polarizable Polar Fluids.- 6.6 Molecular Liquids: Realistic Systems.- 6.6.1 Nitrogen (N2).- 6.6.2 Halogens (Br2, Cl2, I2).- 6.6.3 Benzene (C6H6).- 6.6.4 Naphthalene (C10H8).- 6.6.5 n-Alkanes: CH3(CH2)n?2CH3.- 6.6.6 Water (H2O).- 6.6.7 Methanol (CH3OH).- 6.6.8 Other Polar Systems.- 6.6.9 Mixtures.- 6.7 Solutions.- 6.7.1 Infinite Dilution.- 6.7.2 Finite Concentration.- 6.7.3 Polyelectrolytes and Micelles.- 6.8 Interfaces in Molecular Systems.- 6.8.1 Polar Systems.- (a) Model Systems.- (b) Realistic Systems.- 6.8.2 Chain Molecules Confined by Hard Plates.- References.- 7. Monte Carlo Techniques for Quantum Fluids, Solids and Droplets.- 7.1 Variational Method.- 7.1.1 Variational Wavefunctions.- 7.1.2 The Pair Product Wavefunction.- 7.1.3 Three-Body Correlations.- 7.1.4 Backflow Correlations.- 7.1.5 Pairing Correlations.- 7.1.6 Shadow Wavefunctions.- 7.1.7 Wavefunction Optimisation.- 7.2 Green's Function Monte Carlo and Related Methods.- 7.2.1 Outline of the Method.- 7.2.2 Fermion Methods.- 7.2.3 Shadow Importance Functions.- 7.3 Path Integral Monte Carlo Method.- 7.3.1 PIMC Methodology.- 7.3.2 The High Temperature Density Matrix.- 7.3.3 Monte Carlo Algorithm.- 7.3.4 Simple Metropolis Monte Carlo Method.- 7.3.5 Normal Mode Methods.- 7.3.6 Threading Algorithm.- 7.3.7 Bisection and Staging Methods.- 7.3.8 Sampling Permutations.- 7.3.9 Calculation of the Energy.- 7.3.10 Computation of the Superfluid Density.- 7.3.11 Exchange in Quantum Crystals.- 7.3.12 Comparison of GFMC with PIMC.- 7.3.13 Applications.- 7.4 Some Results for Bulk Helium.- 7.4.1 4He Results.- 7.4.2 3He Results.- 7.4.3 Solid He.- 7.5 Momentum and Related Distributions.- 7.5.1 The Single-Particle Density Matrix.- 7.5.2 y-Scaling.- 7.5.3 Momentum Distribution Results.- 7.6 Droplets and Surfaces.- 7.6.1 Ground States of He Droplets.- 7.6.2 Excitations in Droplets.- 7.6.3 3He Droplets.- 7.6.4 Droplets at Finite Temperature.- 7.6.5 Surfaces and Interfaces.- 7.7 Future Prospects.- References.- 8. Quantum Lattice Problems.- 8.1 Overview.- 8.2 Models.- 8.3 Variational Monte Carlo Method.- 8.3.1 Method and Trial Wavefunctions.- 8.3.2 Results.- 8.4 Green's Function Monte Carlo Method.- 8.4.1 Method.- 8.4.2 Results.- 8.5 Grand Canonical Quantum Monte Carlo Method.- 8.5.1 Method.- 8.5.2 Applications.- 8.6 Projector Quantum Monte Carlo Method.- 8.6.1 Method.- 8.6.2 Applications.- 8.7 Fundamental Difficulties.- 8.7.1 The Sign Problem.- 8.7.2 Numerical Instabilities.- 8.7.3 Dynamic Susceptibilities.- 8.7.4 Applicability.- 8.8 Concluding Remarks.- 8.A Appendix.- References.- 9. Simulations of Macromolecules.- 9.1 Techniques and Models.- 9.1.1 Polymer Models.- (a) Lattice Models.- (b) Off-Lattice Models.- 9.1.2 Monte Carlo Techniques.- (a) Kink-Jump and Crankshaft Algorithm.- (b) Reptation Algorithm.- (c) General Reptation Algorithm.- (d) Grand Canonical Reptation Algorithm.- (e) Collective Reptation Method.- (f) Pivot Algorithm.- (g) Growth and Scanning Algorithms.- 9.2 Amorphous Systems.- 9.2.1 Dynamics of Polymers.- (a) Polymer Melts.- (b) Polymers in Flow.- (c) Gel Electrophoresis.- 9.2.2 The Glassy State.- 9.2.3 Equation of State.- 9.3 Disorder Effects.- 9.3.1 Polymer Chains in Random Media.- 9.3.2 Effect of Disorder on Phase Transitions.- 9.3.3 Diffusion in Disordered Media.- 9.4 Mesomorphic Systems.- 9.4.1 Hard Rods.- 9.4.2 Semirigid Chains.- 9.4.3 Anisotropic Interactions.- 9.5 Networks.- 9.5.1 Tethered Membranes.- 9.5.2 Branched Polymers and Random Networks.- 9.6 Segregation.- 9.6.1 Collapse Transition.- 9.6.2 Polymer Mixtures.- 9.6.3 Dynamics of Decomposition.- 9.7 Surfaces and Interfaces.- 9.7.1 Adsorption on Rough Surfaces.- 9.7.2 Entropic Repulsion.- 9.7.3 Confined Polymer Melts.- 9.8 Special Polymers.- 9.8.1 Polyelectrolytes.- 9.8.2 Proteins.- (a) Protein Folding.- (b) Protein Dynamics.- References.- 10. Percolation, Critical Phenomena in Dilute Magnets, Cellular Automata and Related Problems.- 10.1 Percolation.- 10.2 Dilute Ferromagnets.- 10.3 Cellular Automata.- 10.4 Multispin Programming of Cellular Automata.- 10.5 Kauffman Model and da Silva-Herrmann Algorithm.- References.- 11. Interfaces, Wetting Phenomena, Incommensurate Phases.- 11.1 Interfaces in Ising Models.- 11.1.1 The Three-Dimensional Nearest-Neighbour Ising Model.- 11.1.2 Alloys and Microemulsions.- 11.1.3 Adsorbates and Two-Dimensional Systems.- 11.2 Interfaces in Multistate Models.- 11.3 Dynamical Aspects.- 11.3.1 Growth of Wetting Layers and Interfaces.- 11.3.2 Domain Growth.- 11.4 Spatially Modulated Structures.- 11.5 Conclusions.- References.- 12. Spin Glasses, Orientational Glasses and Random Field Systems.- 12.1 Spin Glasses.- 12.1.1 The Spin Glass Transition.- 12.1.2 The Edwards Anderson Model.- 12.1.3 Phase Transitions.- 12.1.4 The Low Temperature State.- 12.1.5 The Vortex Glass.- 12.2 Potts Glasses.- 12.2.1 Introduction to Potts Glasses.- 12.2.2 Mean-Field Theory.- 12.2.3 The Critical Dimensions.- 12.2.4 The Short-Range Potts Model.- (a) Phenomenological T = 0 Scaling.- (b) Monte Carlo Simulations.- (c) Transfer Matrix Calculations.- (d) High-Temperature Series Expansions.- 12.3 Orientational Glasses.- 12.3.1 Introduction to Orientational Glasses.- 12.3.2 Static and Dynamic Properties of the Isotropic Orientational Glass (m = 3) in Two and Three Dimensions.- 12.3.3 More Realistic Models.- 12.4 The Random-Field Ising Model.- 12.5 Concluding Remarks and Outlook.- References.