Topics in Applied Physics Volume 71 Topics in Applied Physics Founded by Helmut K. V. Lotsch Volume 57 Strong and U1trastrong Magnetic Fields and Their Applications Editor: F. Herlach Volume 58 Hot-Electron Transport in Semiconductors Editor: L. Reggiani Volume 59 Tunable Lasers Editors: L. F. Mollenauer and J. C. White Volume 60 Ultrashort Laser Pulses and Applications Editor: W. Kaiser Volume 61 Photorefractive Materials and Their Applications I Fundamental Phenomena Editors: P. Giinter and J.-P. Huignard Volume 62 Photorefractive Materials and Their Applications II Survey of Applications Editors: P. Giinter and J.-P. Huignard Volume 63 Hydrogen in Intermetallic Compounds I Electronic, Thermodynamic and Crystallographic Properties, Preparation Editor: L. Schlapbach Volume 64 Sputtering by Particle Bombardment III Characteristics of Sputtered Particles, Technical Applications Editors: R. Behrisch and K. Wittmaack Volume 65 Laser Spectroscopy of Solids II Editor: W. M. Yen Volume 66 Light Scattering in Solids V Superlattices and Other Microstructures Editors: M. Cardona and G. Giintherodt Volume 67 Hydrogen in Intermetallic Compounds II Surface and Dynamic Properties, Applications Editor: L. Schlapbach Volume 68 Light Scattering in Solids VI Recent Results, Including High-T Superconductivity c Editors: M. Cardona and G. Giintherodt Volume 69 Unoccupied Electronic States Editors: J. C. Fuggle and J. E. Inglesfield Volume 70 Dye Lasers: 25 Years Editor: M. Stuke Volume 71 The Monte Carlo Method in Condensed Matter Physics Editor: K. Binder Volumes 1-56 are listed on the back inside cover The Monte Carlo Method in Condensed Matter Physics Edited by K. Binder With Contributions by A. Baumgartner K. Binder A. N. Burkitt D. M. Ceperley H. De Raedt A. M. Ferrenberg D. W. Heermann H. 1. Herrmann D. P. Landau D. Levesque W. von der Linden 1. D. Reger K. E. Schmidt W. Selke D. Stauffer R. H. Swendsen 1.-S.Wang 1. 1.Weis A. P. Young With 83 Figures Springer-Verlag Berlin Heidelberg GmbH Professor Dr. Kurt Binder Institut fiir Physik Johannes-Gutenberg-Universităt Mainz Staudingerweg 7 D-6500 Mainz, Fed. Rep. of Germany ISBN 978-3-662-02857-5 Library ofCongress Cataloging-in-Publication Data. The Monte Carlo method in condensed matterphysics/ edited by K.Binder; with contributions byA .Baum giirtner ... [et al.). p. cm. -(Topics in applied physics ; v.71) Inc1udes bibliographical references and index. ISBN 978-3-662-02857-5 ISBN 978-3-662-02855-1 (eBook) DOI 10.1007/978-3-662-02855-1 l.Monte Carl0 method. 2.Statisticalphy- sics. 3. Condensed matter. 1. Binder, K. (Kurt), 1944-. II. Baumgiirtner, A. (Artur) III. Series. QC 174.85.M64M65 1992 530.4'1'01519282-dc20 91-42398 CIP This work is subject to copyright. Ali rights are reserved, whetherthe whole orpart ofthe material is concer ned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, repro duction on microfilm or in other way, and storage in data banks. Duplication of this publication or parts thereofis permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version,and permission foruse must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 UrsprOnglich erschienen bei Springer-Verlag Berlin Heidelberg New York 1992 Softcover reprint of the hardcover 1st edition 1992 The use ofg eneral descriptive narnes, registered narnes, trademarks, etc. in this publication does not imply, even in the absence ofaspecific statement,thatsucbnamesare exemptfrom therelevantprotectivelawsand regulations and therefore free for general use. Typesetting: Thomson Press (India) Ltd., New Delhi; Offsetprinting: Color-Druck Dorli GmbH, Berlin; Binding: Liideritz & Bauer, Berlin 54/3150-5 4 3 2 1 0-Printed on acid-free paper Preface The Monte Carlo method is now widely used and commonly accepted as an important and useful tool in solid state physics and related fields. It is broadly recognized that the technique of "computer simulation" is complementary to both analytical theory and experiment, and can significantly contribute to ad vancing the understanding of various scientific problems. Widespread applications of the Monte Carlo method to various fields of the statistical mechanics of condensed matter physics have already been reviewed in two previously published books, namely Monte Carlo Methods in Statistical Physics (Topics Curro Phys., Vol. 7, 1st edn. 1979, 2ndedn. 1986) and Applications oft he Monte Carlo Method in Statistical Physics (Topics Curro Phys., Vol. 36, 1st edn. 1984, 2nd edn. 1987). Meanwhile the field has continued its rapid growth and expansion, and applications to new fields have appeared that were not treated at all in the above two books (e.g. studies of irreversible growth phenomena, cellular automata, interfaces, and quantum problems on lattices). Also, new methodic aspects have emerged, such as aspects of efficient use of vector com puters or parallel computers, more efficient analysis of simulated systems con figurations, and methods to reduce critical slowing down at i>hase transitions. Taken together with the extensive activity in certain traditional areas of research (simulation of classical and quantum fluids, of macromolecular materials, of spin glasses and quadrupolar glasses, etc.), there is clearly a need for a new book complementing the previous volumes. Thus, in the present book we present selected state-of-the-art reviews of those fields which have seen dramatic progress during the last couple of years and provide the reader with an up-to-date guide to the' explosively growing original literature. Although the present book contains several thousand references, it must be realized that by now it has become impossi ble to describe all the research that uses Monte Carlo methods in condensed matter physics! We apologize to all colleagues whose work has been only briefly mentioned, or even not quoted at all, for the need to make a selection where certain topics had to be emphasized more than others. We hope that nevertheless the present book constitutes a useful landmark in a rapidly evolving field. Again, it is a great pleasure to thank the team of expert authors that contributed to the present book for their coherent and constant efforts and their fruitful collabo ration. Mainz, December 1991 Kurt Binder Contents 1. Introduction By K. Binder (With 9 Figures) . . . . . . . . . . . . . . . . . . . . .. . 1 1.1 General Remarks .......................... . 1 1.2 Progress in the Understanding of Finite Size Effects at Phase Transitions ........................ . 4 1.2.1 Asymmetric First-Order Phase Transition ........ . 4 1.2.2 Coexisting Phases . . . . . . . . . . . . . . . . . . . 9 1.2.3 Critical Phenomena Studies in the Microcanonical Ensemble ......... . 10 1.2.4 Anisotropy Effects in Finite Size Scaling . . . . . . . . . . 13 1.3 Statistical Errors . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References .......... . 19 2. Vectorisation of Monte Carlo Programs for Lattice Models Using Supercomputers By D.P. Landau (With 11 Figures) .................. .. 23 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Technical Details .......................... . 24 2.2.1 Basic Principles ....................... . 24 2.2.2 Some "Dos" and "Don'ts" of Vectorisation ....... . 26 2.3 Simple Vectorisation Algorithms ................. . 27 2.4 Vectorised Multispin Coding Algorithms ... . 28 2.5 Vectorised Multilattice Coding Algorithms ........ . 32 2.6 Vectorised Microcanonical Algorithms .......... . 34 2.7 Some Recent Results from Vectorised Algorithms .... . 35 2.7.1 Ising Model Critical Behaviour ........... . 35 2.7.2 First-Order Transitions in Potts Models ......... . 36 2.7.3 Dynamic Critical Behaviour ................ . 37 2.7.4 Surface and Interface Phase Transitions ..... . 39 2.7.5 Bulk Critical Behaviour in Classical Spin Systems 42 2.7.6 Quantum Spin Systems ............... . 43 2.7.7 Spin Exchange and Diffusion ........... . 45 2.7.8 Impurity Systems ...................... . 46 2.7.9 Other Studies ........................ . 47 2.8 Conclusion .. 49 References ................................. . 49 VIII Contents 3. Parallel Algorithms for Statistical Physics Problems By D.W. Heermann and A.N. Burkitt (With 8 Figures) ........ 53 3.1 Paradigms of Parallel Computing ................. 54 3.1.1 Physics-Based Description . . . . . . . . . . . . . . . . .. 54 (a) Event Parallelism . . . . . . . . . . . . . . . . . . . .. 55 (b) Geometric Parallelism ......... . . . . . . . .. 56 (c) Algorithmic Parallelism ................. 57 3.1.2 Machine-Based Description . . . . . . . . . . . . . . . .. 57 (a) SIMD Architecture . . . . . . . . . . . . . . . . . . .. 57 (b) MIMD Architecture ................... 58 (c) The Connectivity .......... . . . . . . . . . .. 58 (d) Measurements of Machine Performance ....... 59 3.2 Applications on Fine-Grained SIMD Machines ......... 62 3.2.1 Spin Systems. . . . . . . . . . . . . . . . . . . . . . . . .. 62 3.2.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . .. 63 3.3 Applications on Coarse-Grained MIMD Machines ....... 64 3.3.1 Molecular Dynamics ..... . . . . . . . . . . . . . . .. 64 3.3.2 Cluster Algorithms for the Ising Model .......... 66 3.3.3 Data Parallel Algorithms .................. 69 (a) Long-Range Interactions. . . . . . . . . . . . . . . .. 70 (b) Polymers. . . . . . . . . . . . . . . . . . . . . . . . .. 70 3.4 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72 References .................................. 73 4. New M.onte Carlo Methods for Improved Efficiency of Computer Simulations in Statistical Mechanics By R.H. Swendsen, 1.-S. Wang and A.M. Ferrenberg . . . . . . . .. 75 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75 4.2 Acceleration Algorithms ....................... 76 4.2.1 Critical Slowing Down and Standard Monte Carlo Method ............ 76 4.2.2 Fortuin-Kasteleyn Transformation ............ 77 4.2.3 Swendsen-Wang Algorithm . . . . . . . . . . . . . . . .. 78 4.2.4 Further Developments ................ . . .. 80 4.2.5 Replica Monte Carlo Method . . . . . . . . . . . . . . .. 82 4.2.6 Multigrid Monte Carlo Method .............. 83 4.3 Histogram Methods ......................... 84 4.3.1 The Single-Histogram Method ............... 84 4.3.2 The Multiple-Histogram Method . . . . . . . . . . . . .. 85 4.3.3 History and Applications .................. 87 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89 References .............................. 89 5. Simulation of Random Growth Processes By H.J. Herrmann (With 17 Figures) . . . . . . . . . . . . . . . . . .. 93 5.1 Irreversible Growth of Clusters .... . . . . . . . . . . . . . .. 93 Contents IX 5.1.1 A Simple Example of Cluster Growth: The Eden Model ................ . 93 5.1.2 Laplacian Growth ................. . 96 (a) Moving Boundary Condition Problems ... . 96 (b) Numerical Simulation of Dielectric Breakdown and DLA ......................... . 97 (c) Fracture ......................... . 101 5.2 Reversible Probabilistic Growth ................. . 103 5.2.1 Cellular Automata ..... ................ . 103 5.2.2 Damage Spreading in the Monte Carlo Method .. . 104 5.2.3 Numerical Results for the Ising Model .......... . 105 5.2.4 Heat Bath Versus Glauber Dynamics in the Ising Model ...................... . 107 5.2.5 Relationship Between Damage and Thermodynamic Properties . . . . . . . . . . .. . . . 108 5.2.6 Damage Clusters ...................... . 112 5.2.7 Damage in Spin Glasses .................. . 114 5.2.8 More About Damage Spreading ........... .. . 117 5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . 117 References ........................ . 118 6. Recent Progress in the Simulation of Classical Fluids By D. Levesque and J.J. Weis ...................... . 121 6.1 Improvements of the Monte Carlo Method . . . . . . . . . . . . 121 6.1.1 Metropolis Algorithm ................... . 121 6.1.2 Monte Carlo Simulations and Statistical Ensembles ................. . 123 (a) Canonical, Grand Canonical and Semi-grand Ensembles ......... . . . . . . 123 (b) Gibbs Ensemble ..................... . 124 (c) MC Algorithm for "Adhesive" Particles ....... . 126 6.1.3 Monte Carlo Computation of the Chemical Potential and the Free Energy .................... . 126 (a) Chemical Potential . . . . . . . . . . . . . . . . . . . . 126 (b) Free Energy ....................... . 129 6.1.4 Algorithms for Coulombic and Dielectric Fluids .... . 130 6.2 Pure Phases and Mixtures of Simple Fluids . . . . . . . . . . . . 132 6.2.1 Two-Dimensional Simple Fluids ............. . 132 6.2.2 Three-Dimensional Monatomic Fluids .......... . 134 6.2.3 Lennard-Jones Fluids and Similar Systems ....... . 136 6.2.4 Real Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.2.5 Mixtures of Simple Fluids .......... ....... . 140 (a) Hard Core Systems . . . . . . . . . . ......... . 140 (b) LJ Mixtures ....................... . 141 (c) Polydisperse Fluids . . . . . . . . ........... . 143 6.3 Coulombic and Ionic Fluids . . . . . . . . . . . . . . . . . . . . . 144 X Contents 6.3.1 One-Component Plasma, Two-Component Plasma and Primitive Models of Electrolyte Solutions 144 (a) OCP and TCP ................... . 144 (b) Primitive Models .................. . 145 6.3.2 Realistic Ionic Systems . . . . . . . . .. 147 6.4 Simulations of Inhomogeneous Simple Fluids 149 6.4.1 Liquid-Vapour Interfaces .. 149 6.4.2 Fluid-Solid Interfaces .... 150 6.4.3 Interfaces of Charged Systems 153 6.4.4 Fluids in Narrow Pores .. 155 6.5 Molecular Liquids: Model Systems . 157 6.5.1 Two-Dimensional Systems .. 157 6.5.2 Convex Molecules (Three-Dimensional) .... 158 (a) Virial Coefficients and the Equation of State . . ... 159 (b) Pair Distribution Function .. .. 160 (c) Phase Transitions ......... . 161 6.5.3 Site-Site Potentials . . . . . . . . . . . 163 6.5.4 Chain Molecules ...... . 164 6.5.5 Dipolar Systems . . . . . . . . . . . .. 165 6.5.6 Quadrupolar Systems . . . . . . .... 167 6.5.7 Polarizable Polar Fluids . . . . . . . . 167 6.6 Molecular Liquids: Realistic Systems . . . . . . . . . . . . . . .. 168 6.6.1 Nitrogen (N 2) .. ... 169 6.6.2 Halogens (Br 2, C12, 12) ................... . 169 6.6.3 Benzene (C6H6) •••....•.••.•...•••.•.•• 170 6.6.4 Naphthalene (ClOHg) .................... . 170 6.6.5 n-Alkanes: CH3(CH2)n-2CH3 .. 171 6.6.6 Water (H20) .............. . 172 6.6.7 Methanol (CH30H) ......... . 177 6.6.8 Other Polar Systems . 178 6.6.9 Mixtures ....................... .... . 178 6.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.7.1 Infinite Dilution ................. . 179 6.7.2 Finite Concentration .............. . 181 6.7.3 Polyelectrolytes and Micelles ......... . 183 6.8 Interfaces in Molecular Systems . . . . . . . . . . . . . . . . . . . 185 6.8.1 Polar Systems ................. . 185 (a) Model Systems ............... . 186 (b) Realistic Systems .............. . 187 6.8.2 Chain Molecules Confined by Hard Plates 190 References ........................... . 191 7. Monte Carlo Techniques for Quantum Fluids, Solids and Droplets By K.E. Schmidt and D.M. Ceperley (With 12 Figures) ........ 205 7.1 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . .. 207 Contents XI 7.1.1 Variational Wavefunctions ................. 207 7.1.2 The Pair Product Wavefunction .............. 207 7.1.3 Three-Body Correlations . . . . . . . . . . . . . . . . . .. 209 7.1.4 Backflow Correlations .................... 210 7.1.5 Pairing Correlations ..................... 211 7.1.6 Shadow Wavefunctions ................... 212 7.1.7 Wavefunction Optimisation . . . . . . . . . . . . . . . .. 214 7.2 Green's Function Monte Carlo and Related Methods . . . . .. 215 7.2.1 Outline of the Method .................... 215 7.2.2 Fermion Methods . . . . . . . . . . . . . . . . . . . . . .. 216 7.2.3 Shadow Importance Functions ............... 217 7.3 Path Integral Monte Carlo Method ................ 218 7.3.1 PIMC Methodology ..................... 218 7.3.2 The High Temperature Density Matrix .......... 219 7.3.3 Monte Carlo Algorithm ................... 221 7.3.4 Simple Metropolis Monte Carlo Method ......... 221 7.3.5 Normal Mode Methods ................... 222 7.3.6 Threading Algorithm . . . . . . . . . . . . . . . . . . . .. 222 7.3.7 Bisection and Staging Methods . . . . . . . . . . . . . .. 222 7.3.8 Sampling Permutations ................... 224 7.3.9 Calculation of the Energy .................. 225 7.3.10 Computation of the Superfluid Density .......... 226 7.3.11 Exchange in Quantum Crystals . . . . . . . . . . . . . .. 227 7.3.12 Comparison of GFMC with PIMC . . . . . . . . . . . .. 229 7.3.13 Applications ........ . . . . . . . . . . . . . . . . .. 230 7.4 Some Results for Bulk Helium ................... 230 7.4.1 4He Results .......................... 230 7.4.2 3He Results .......................... 233 7.4.3 Solid He ............................ 234 7.5 Momentum and Related Distributions. . . . . . . . . . . . . .. 234 7.5.1 The Single-Particle Density Matrix . . . . . . . . . . . .. 234 7.5.2 y-Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . .. 236 7.5.3 Momentum Distribution Results ... . . . . . . . . . .. 237 7.6 Droplets and Surfaces ........................ 240 7.6.1 Ground States of He Droplets . . . . . . . . . . . . . . .. 240 7.6.2 Excitations in Droplets . . . . . . . . . . . . . . . . . . .. 242 7.6.3 3He Droplets . . . . . . . . . . . . . . . . . . . . . . . . .. 242 7.6.4 Droplets at Finite Temperature . . . . . . . . . . . . . .. 243 7.6.5 Surfaces and Interfaces . . . . . . . . . . . . . . . . . . .. 243 7.7 Future Prospects ........................... 244 References .................................. 245 8. Quantum Lattice Problems By H. De Raedt and W. von der Linden (With 1 Figure) . . . . . .. 249 8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 249