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Interaction Range, Universality and the Upper Critical Dimension

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Interaction Range, Universality and the Upper Critical Dimension Interaction Range, Universality and the Upper Critical Dimension Interaction Range, Universality and the Upper Critical Dimension PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus Prof. dr. ir. J. Blaauwendraad, in het openbaar te verdedigen ten overstaan van een commissie, door het College van Dekanen aangewezen, op dinsdag 9 december 1997 te 13.30 uur door Erik LUIJTEN doctorandus in de natuurkunde geboren te Nijmegen Dit proefschrift is goedgekeurd door de promotoren: Prof. dr. H. W. J. Blote¨ Prof.dr.ir.J.J.J.Kokkedee Samenstelling promotiecommissie: Rector Magnificus, voorzitter Prof. dr. H. W. J. Blote,¨ Rijksuniversiteit Leiden, promotor Prof. dr. ir. J. J. J. Kokkedee, Technische Universiteit Delft, promotor Prof. dr. H. van Beijeren, Universiteit Utrecht Prof. dr. K. Binder, Johannes Gutenberg-Universitat¨ Mainz Prof. dr. S. W. de Leeuw, Technische Universiteit Delft Prof. dr. J. M. J. van Leeuwen, Rijksuniversiteit Leiden Published and distributed by: Delft University Press Mekelweg 4 2628 CD Delft The Netherlands Telephone: +31 15 2783254 Fax: +31 15 2781661 E-mail: [email protected] ISBN 90-407-1552-1 / CIP Copyright c 1997 by E. Luijten All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval sys- tem, without permission from the publisher: Delft University Press. Printed in the Netherlands Contents 1 Introduction and outline 1 2 Monte Carlo algorithm for spin models with long-range interactions 7 2.1 Introduction............................. 7 2.2 Clustermethods........................... 8 2.3 Building clusters in systems with long-range interactions . 10 2.3.1 Long-rangeHamiltonian................... 10 2.3.2 Look-uptable........................ 12 2.3.3 Continuousbondprobability................ 12 2.3.4 Comparisontoconventionalclusteralgorithms....... 15 2.4 A“trivial”example:themean-fieldmodel.............. 16 2.5 Generalizations............................ 19 References.................................. 20 3 The Ising model in three dimensions 23 3.1 Introduction............................. 23 3.2 Modelsandalgorithms........................ 25 3.3 Randomnumbers........................... 28 3.4 Atestofuniversality......................... 28 3.5 Determinationoftherenormalizationexponents.......... 33 3.5.1 Theenergy.......................... 34 3.5.2 Thespecificheat....................... 35 3.5.3 Thespin–spincorrelationfunction............. 36 3.5.4 Themagneticsusceptibility................. 38 3.5.5 The temperature derivative of χ ............... 40 3.5.6 The temperature derivative of Q ............... 41 3.6 Simultaneousfitsforthethreemodels................ 42 3.7 Discussionandconclusion...................... 44 References.................................. 46 v vi CONTENTS 4 Critical behaviour of spin models with algebraically decaying interactions I At and above the upper critical dimension 49 4.1 Introduction............................. 49 4.2 Rigorousresultsfortheone-dimensionalcase............ 51 4.3 Renormalization-groupstudyofthecriticalbehaviour....... 52 4.4 Numericalresultsandcomparisonwithearlierresults....... 60 4.4.1 Simulations.......................... 60 4.4.2 Determination of the critical temperatures, the amplitude ratio Q andthethermalexponent.............. 61 4.4.3 Determinationofcriticalexponents............. 70 4.5 Conclusions.............................. 76 References.................................. 77 5 Critical behaviour of spin models with algebraically decaying interactionsII Below the upper critical dimension 79 5.1 Introduction............................. 79 5.2 Renormalization-group study of the nonclassical regime . 80 5.3 The1DIsingmodelwithlong-rangeinteractions.......... 90 5.4 The2DIsingmodelwithlong-rangeinteractions.......... 96 5.4.1 Introduction......................... 96 5.4.2 Regime A: 1 <σ<7/4 ................... 96 5.4.3 Regime B: 7/4 ≤ σ ≤ 2 ................... 96 5.4.4 Regime C: σ>2....................... 99 5.4.5 Discussionandconclusion.................. 100 5.5 The3DIsingmodelwithlong-rangeinteractions.......... 102 5.5.1 Introduction......................... 102 5.5.2 Systems with a decay parameter σ = 1.6, 1.8, 2.0 . 102 5.5.3 Systems with a decay parameter σ = 2.2, 2.5, 3.0 . 103 5.5.4 Discussionandconclusions................. 104 5.6 Conclusion.............................. 104 References.................................. 105 6 The Ising model in four and five dimensions 107 6.1 Introduction............................. 107 6.2 Theory................................ 110 6.3 Thespecificheatofthemean-fieldmodel.............. 115 6.4 Numerical results for the case d = 5................. 117 6.4.1 Generalconsiderations.................... 117 6.4.2 The amplitude ratio Q .................... 117 6.4.3 Themagneticsusceptibility................. 120 6.4.4 Thespin–spincorrelationfunction............. 120 CONTENTS vii 6.4.5 Thefourthpowerofthemagnetizationdensity....... 121 6.4.6 Theenergy.......................... 122 6.4.7 Thespecificheat....................... 123 6.4.8 The first temperature derivative of the specific heat . 124 6.4.9 The second temperature derivative of the specific heat . 124 6.4.10 The temperature derivative of the magnetic susceptibility . 126 6.4.11 The temperature derivative of Q ............... 127 6.5 Numerical results for the case d = 4................. 127 6.5.1 Generalconsiderations.................... 127 6.5.2 The amplitude ratio Q .................... 128 6.5.3 Themagneticsusceptibility................. 130 6.5.4 Thespin–spincorrelationfunction............. 131 6.5.5 Thefourthpowerofthemagnetizationdensity....... 132 6.5.6 Theenergy.......................... 133 6.5.7 Thespecificheat....................... 134 6.5.8 The first temperature derivative of the specific heat . 135 6.5.9 The second temperature derivative of the specific heat . 136 6.5.10 The temperature derivative of the magnetic susceptibility . 137 6.5.11 The temperature derivative of Q ............... 137 6.6 Discussionandconclusions..................... 138 6.6.1 Thefive-dimensionalIsingmodel.............. 138 6.6.2 Thefour-dimensionalIsingmodel............. 142 References.................................. 144 7 Crossover from Ising to classical critical behaviour in two-dimensional systems 147 7.1 Introduction............................. 147 7.2 Rangedependenceofcriticalamplitudes............... 149 7.3 MonteCarloresults.......................... 156 7.3.1 Definitionofthemodel................... 156 7.3.2 Determinationofthecriticaltemperature.......... 157 7.3.3 Range dependence of the magnetization density . 161 7.3.4 Rangedependenceofthesusceptibility........... 162 7.3.5 Spin–spincorrelationfunction............... 163 7.4 Crossoverscaling........................... 166 7.5 Finite-sizecrossoverscaling..................... 167 7.5.1 Generalconsiderations.................... 167 7.5.2 Absolutemagnetizationdensity............... 168 7.5.3 Magneticsusceptibility.................... 171 7.5.4 Spin–spincorrelationfunction............... 172 7.5.5 Universalamplituderatio.................. 172 viii CONTENTS 7.6 Thermalcrossoverscaling...................... 176 7.6.1 Generalconsiderations.................... 176 7.6.2 Absolutemagnetizationdensity............... 177 7.6.3 Magneticsusceptibility.................... 182 7.7 EVectiveexponents.......................... 188 7.8 Conclusions.............................. 192 References.................................. 193 A Some exact calculations for the mean-field model 195 B Fourier transform of a spherically shaped interaction profile 199 Summary 201 Samenvatting 205 Acknowledgements 209 Curriculum Vitae 211 List of publications 213 Chapter 1 Introduction and outline Among the most challenging problems in theoretical physics are those which involve a large number of coupled degrees of freedom. In such problems, the macroscopic properties of a system are determined by microscopic behaviour on many diVerent length scales, all of which must be taken into account for the solution of the prob- lem. Examples are quantum electrodynamics, turbulence and critical phenomena. The latter subject deals with the behaviour of systems near a critical point. From every-day experience, we know that substances can appear in diVerent phases; e.g., water can be a solid, a liquid or a gas. Transitions between these phases can be in- duced by changing the temperature of the substance or the pressure under which it is placed. Starting in the gas phase and increasing the pressure while keeping the tem- perature fixed at some (suYciently low) value, the substance will at a certain pres- sure go over to the solid or to the liquid phase. This transition is marked by a sudden change (a discontinuity) in the density of the substance. However, if we repeat the same experiment at a higher temperature, the jump in the density will be smaller, and at some temperature it will even vanish. At this critical point (i.e., at this partic- ular temperature and pressure) the distinction between the gas and liquid phase has completely disappeared. Here, some remarkable aspects arise: despite the fact that the fundamental interactions are short-ranged, they cooperate in such a way that density fluctuations occur at all length scales. The correlation length, which mea- sures the size of the largest fluctuations in the system,
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