Empirical relations between static and dynamic exponents for Ising model cluster algorithms.

Paul D Coddington,Clive F Baillie

doi:10.1103/physrevlett.68.962

Abstract

We have measured the autocorrelations for the Swendsen-Wang and the Wolff cluster update algorithms for the Ising model in two, three, and four dimensions. The data for the Wolff algorithm suggest that the autocorrelations are linearly related to the specific heat, in which case the dynamic critical exponent is ${\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$${\mathit{E}}^{\mathrm{W}}$=\ensuremath{\alpha}/\ensuremath{\nu}. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that ${\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$${\mathit{E}}^{\mathrm{SW}}$=\ensuremath{\beta}/\ensuremath{\nu} for the Ising model.

Highlights

The Monte Carlo cluster update algorithms of Swendsen and Wang (SW) [1] and Wolff [2] can dramatically reduce critical slowing down in computer simulations of spin models, and greatly increase the computational efficiency of the simulations
Since this scaling ratio is an estimator for the susceptibility [2], the dynamic critical exponent z for the unscaled autocorrelations is given by z = z + (d − γ/ν), where ν is the critical exponent for the correlation length, and γ is the critical exponent for the susceptibility, which diverges as Lγ/ν
For the Wolff algorithm in all dimensions, and the SW algorithm in two dimensions, it is very difficult to distinguish between a small exponent and a logarithmic increase in the autocorrelations

Summary

Introduction

The Monte Carlo cluster update algorithms of Swendsen and Wang (SW) [1] and Wolff [2] can dramatically reduce critical slowing down in computer simulations of spin models, and greatly increase the computational efficiency of the simulations (for reviews of cluster algorithms, see refs. [3] [4] ). There is no known theory which can predict the value of the dynamic critical exponent z for any spin model, a rigorous bound on z for the SW algorithm for Potts models has been derived [5] Another problem which is not well understood is why the SW and Wolff algorithms give similar values of z for the 2-d Potts model [6] , but have very different behavior for other models, such as the Ising model in more than two dimensions [7] [8]. The measurement of dynamic critical exponents is notoriously difficult, and both very good statistics and very large lattices are required in order to obtain accurate results. This is certainly the case for the Ising model, where a number of different measurements have given conflicting results. The mean-field data are consistent with z being 0 for the Wolff algorithm [8] and 1 for SW [12] , with the latter result being supported by theoretical arguments

Simulations

Results

Conclusions