University of Rhode Island DigitalCommons@URI Physics Faculty Publications Physics 2000 Monte Carlo Computation of Correlation Times of Independent Relaxation Modes at Criticality M. P. Nightingale University of Rhode Island, [email protected] H. W.J. Blöte Follow this and additional works at: https://digitalcommons.uri.edu/phys_facpubs Terms of Use All rights reserved under copyright. Citation/Publisher Attribution Nightingale, M. P., & Blöte, H. W.J. (2000). Monte Carlo computation of correlation times of independent relaxation modes at criticality. Physical Review B, 62(2), 1089-1101. doi: 10.1103/PhysRevB.62.1089 Available at: http://dx.doi.org/10.1103/PhysRevB.62.1089 This Article is brought to you for free and open access by the Physics at DigitalCommons@URI. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. PHYSICAL REVIEW B VOLUME 62, NUMBER 2 1 JULY 2000-II Monte Carlo computation of correlation times of independent relaxation modes at criticality M. P. Nightingale Department of Physics, University of Rhode Island, Kingston, Rhode Island 02881 H. W. J. Blo¨te Faculty of Applied Sciences, Delft University, P.O. Box 5046, 2600 GA Delft, The Netherlands Lorentz Institute, Leiden University, Niels Bohrweg 2, P.O. Box 9506, 2300 RA Leiden, The Netherlands ͑Received 13 January 2000͒ We investigate aspects of the universality of Glauber critical dynamics in two dimensions. We compute the critical exponent z and numerically corroborate its universality for three different models in the static Ising universality class and for five independent relaxation modes. We also present evidence for universality of amplitude ratios, which shows that, as far as dynamic behavior is concerned, each model in a given universality class is characterized by a single nonuniversal metric factor which determines the overall time scale. This paper also discusses in detail the variational and projection methods that are used to compute relaxation times with high accuracy. I. INTRODUCTION dates has been elusive. This is caused by the difficulty of obtaining the required accuracy in estimates of the dynamic Critical-point behavior is a manifestation of power-law critical exponent. Under these circumstances it is evident that divergences of the correlation length and the correlation only a limited progress has been made with respect to the time. The power laws that describe the divergence of the interesting questions regarding dynamic universality classes. correlation length on approach of the critical point are ex- In this paper we present a detailed exposition of a method 6,7 pressed by means of critical exponents that are dependent on of computing dynamic exponents with high accuracy. We the direction of this approach, which may, e.g., be ordering- consider single spin-flip Glauber dynamics. This is defined field like or temperature like. The exponents describing the by a Markov matrix, and computation of the correlation time singularities in thermodynamic quantities can be expressed in is viewed here as an eigenvalue problem, since correlation terms of the same exponents. In addition to the exponents times can be obtained from the subdominant eigenvalues of defining these power laws, another critical exponent, viz., the the Markov matrix. dynamic exponent z, is required for the singularities in the If a thermodynamic system is perturbed out of equilib- dynamics. This exponent z is defined by the relationship that rium, different thermodynamic quantities relax back at a dif- holds between the correlation length ␰ and the correlation ferent rates. More generally, there are infinitely many inde- time ␶, namely, ␶ϰ␰z. pendent relaxation modes for a system in the thermodynamic One of the directions along which one can approach the limit. Let us label the models within a given universality ␬ ␶ critical singularity is the finite-size direction; i.e., one in- class by means of , and denote by Li␬ the autocorrelation creases the system size L while keeping the independent time of relaxation mode i of a system of linear dimension L. thermodynamic variables at their infinite-system critical val- In this paper we present strong numerical evidence that, as ues. In this case, ␰ϰL so that ␶ϰLz. This relation has been indeed renormalization group theory suggests, at criticality used extensively to obtain the dynamic exponent z from the relaxation times have the following factorization prop- finite-size calculations. erty: In this paper we deal with universality of dynamic ␶ Ϸ z ͑ ͒ critical-point behavior. One would not expect systems in dif- Li␬ m␬AiL , 1 ferent static universality classes to have the same dynamic exponents, and even within the same static universality class, where m␬ is a nonuniversal metric factor, which differs for different dynamics may have different exponents. For in- different representatives of the same universality class as in- stance, in the case of the Ising model, Kawasaki dynamics dicated; Ai is a universal amplitude, which depends on the which satisfies a local conservation law1 has a larger value of mode i; and z is the universal dynamical exponent introduced z than Glauber dynamics,2 in which such a conservation law above. is absent. Also the introduction of nonlocal spin updates, as While the relaxation time of the slowest relaxation mode realized, e.g., in cluster algorithms, is known to lead to a is obtained from the second-largest eigenvalue of the Mar- different dynamic universal behavior.3–5 kov matrix, lower-lying eigenvalues yield the relaxation Conservation laws and nonlocal updates tend to have a times of faster modes. To compute these we construct, em- large effect on the numerical value of the dynamic expo- ploying a Monte Carlo method, variational approximants for nents, but until fairly recently, numerical resolution of the several eigenvectors. These approximants are called opti- expected differences of dynamic exponents of systems in dif- mized trial vectors. The corresponding eigenvalues can then ferent static universality classes for dynamics with local up- be estimated by evaluating with Monte Carlo techniques the 0163-1829/2000/62͑2͒/1089͑13͒/$15.00PRB 62 1089 ©2000 The American Physical Society 1090 M. P. NIGHTINGALE AND H. W. J. BLO¨ TE PRB 62 Ϯ overlap of these trial vectors and the corresponding matrix si ,...,sl assume values 1. Periodic boundaries are used elements of the Markov matrix in the truncated basis throughout. In particular, we focus on models described by spanned by these optimized trial vectors. It should be noted three ratios ␤ϭKЈ/K, namely, ␤ϭϪ1/4, 0 ͑nearest- that both the optimization scheme and the evaluation of these neighbor model͒ and 1 ͑equivalent-neighbor model͒. The matrix elements critically depend on the fact that the Markov nonplanar models for ␤Þ0 are not exactly solvable and their matrix is sparse. That is, the number of configurations acces- critical points are known only approximately. Yet it was sible from any given configuration is equal to the number of demonstrated to a high degree of numerical accuracy that sites only, rather than the number of possible spin configu- that they belong to the static Ising universality class.11,12 For rations. the nearest-neighbor model the critical coupling is K ϭ 1 ϩͱ Given such fixed trial vectors, this approach has the ad- 2 ln(1 2); for the other two models estimates of the vantage of simplicity and high statistical accuracy, but the critical points are Kϭ0.190 192 680 7(2) for ␤ϭ1 and K disadvantage is that results are subject to systematic, varia- ϭ0.697 220 7(2) for ␤ϭϪ1/4.12,13 tional errors, which only vanish in the ideal limit where the We use the dynamics of the heat-bath algorithm with ran- variational vectors become exact eigenvectors or span an in- dom site selection. The single-spin-flip dynamics is deter- variant subspace of the Markov matrix. Since the condition mined by the Markov matrix P defined as follows. The ele- is rarely satisfied in cases of practical interest, a projection ment P(SЈ,S) is the transition probability of going from Monte Carlo method is then used, to reduce the systematic configuration S to SЈ.IfS and SЈ differ by more than one error, but this is at the expense of an increase of the statisti- spin, P(SЈ,S)ϭ0. If both configurations differ by precisely cal errors. The method we use in this paper is a combination one spin, and generalization of the work of Umrigar et al.8 and that of 9 Ceperley and Bernu. 1 H͑SЈ͒ϪH͑S͒ To summarize, the Monte Carlo method discussed here P͑SЈ,S͒ϭ ͭ 1Ϫtanhͫ ͬͮ , ͑3͒ 2 2kT consists of two phases. In the first phase, trial vectors are 2L optimized. The ultimate, yet unattainable goal of this phase where L2 is the total number of spins. The diagonal elements is to construct exact eigenvectors. In this phase of the com- P(S,S) follow from the conservation of probability, putation, very small Monte Carlo samples are used, consist- ing typically of no more than a few thousand spin configu- ͑ Ј ͒ϭ ͑ ͒ rations. In the second phase, one performs a standard Monte ͚ P S ,S 1, 4 Ј Carlo computation in which one reduces statistical errors by S 2 increasing the length of the computation rather than the qual- where SЈ runs over all possible 2L spin configurations. ity of the variational approximation. We denote the probability of finding spin configuration S ␳ The computed correlation times, derived from the partial at time t by t(S). By design, the stationary state of the solution of the eigenvalue problem as sketched above, are Markov process is the equilibrium distribution used in a finite-size analysis to compute the dynamic critical ϪH ͒ ␺ ͒2 exponent z.