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Dynamic critical behavio(u)r of a cluster algorithm for the Ashkin-Teller model

Salas, J, Alan D. Sokal

Published Sep 12, 1995• 2 citations
OpenAlex

Abstract

We study the dynamic critical behavior of a Swendsen--Wang--type algorithm for the Ashkin--Teller model. We find that the Li--Sokal bound on the autocorrelation time (\\tau_{{\\rm int},{\\cal E}} \\ge {\\rm const} \\times C_H) holds along the self-dual curve of the symmetric Ashkin--Teller model, but this bound is apparently not sharp. The ratio \\tau_{{\\rm int},{\\cal E}}/C_H appears to tend to infinity either as a logarithm or as a small power (0.05 \\ltapprox p \\ltapprox 0.12).

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Physical Review Letters•Feb 17, 1992•45 citations
We study the dynamic critical behavior of a Swendsen--Wang--type algorithm for the Ashkin--Teller model. We find that the Li--Sokal bound on the autocorrelation time (\tau_{{\rm int},{\cal E}} \ge {\rm const} \times C_H) holds along the self-dual curve of the symmetric Ashkin--Teller model, but this bound is apparently not sharp. The ratio \tau_{{\rm int},{\cal E}}/C_H appears to tend to infinity either as a logarithm or as a small power (0.05 \ltapprox p \ltapprox 0.12).
We study the dynamic critical behavior of two algorithms: the Swendsen-Wang algorithm for the two-dimensional Potts model with q=2,3,4 and a Swendsen-Wang-type algorithm for the two-dimensional symmetric Ashkin-Teller model on the self-dual curve. We find that the Li--Sokal bound on the autocorrelation time τ_{{\rm int},{\cal E}} \geq const \times C_H is almost, but not quite sharp. The ratio τ_{{\rm int},{\cal E}}/C_H appears to tend to infinity either as a logarithm or as a small power (0.05 \ltapprox p \ltapprox 0.12). We also show that the exponential autocorrelation time τ_{{\rm exp},{\cal E}} is proportional to the integrated autocorrelation time τ_{{\rm int},{\cal E}}.
A cluster Monte Carlo algorithm for the Ashkin-Teller (AT) model is constructed according to the guidelines of a general scheme for such algorithms. Its dynamical behavior is tested for the square-lattice AT model. We perform simulations on the line of critical points along which the exponents vary continuously, and find that critical slowing down is significantly reduced. We find continuous variation of the dynamical exponent z along the line, following the variation of the ratio of specific-heat and correlation-length exponents \ensuremath{\alpha}/\ensuremath{\nu}, in a manner which satisfies the Li-Sokal bound zcluster{\mathit{z}}_{\mathrm{cluster}}zcluster​\ensuremath{\ge}\ensuremath{\alpha}/\ensuremath{\nu}, that was so far proved only for Potts models.
Simulations of the two-dimensional Ising and three-state Potts models at their critical points are performed using the invaded cluster (IC) algorithm. It is argued that observables measured on a sublattice of size l should exhibit a crossover to Swendsen-Wang (SW) behavior for l sufficiently less than the lattice size L, and a scaling form is proposed to describe the crossover phenomenon. It is found that the energy autocorrelation time ${\ensuremath{\tau}}_{\ensuremath{\varepsilon}}(l,L)$ for an $l\ifmmode\times\else\texttimes\fi{}l$ sublattice attains a maximum in the crossover region, and a dynamic exponent zIC{z}^{\mathrm{IC}}zIC for the IC algorithm is defined according to ${\ensuremath{\tau}}_{\ensuremath{\varepsilon},\mathrm{max}}\ensuremath{\sim}{L}^{{z}^{\mathrm{IC}}}.$ Simulation results for the three-state model yield ${z}^{\mathrm{IC}}=0.346\ifmmode\pm\else\textpm\fi{}0.002,$ which is smaller than values of the dynamic exponent found for the SW and Wolff algorithms and also less than the Li-Sokal bound. The results are less conclusive for the Ising model, but it appears that zIC<0.21{z}^{\mathrm{IC}}<0.21zIC<0.21 and possibly that ${\ensuremath{\tau}}_{\ensuremath{\varepsilon},\mathrm{max}}\ensuremath{\sim}\mathrm{ln}L$ so that zIC=0{z}^{\mathrm{IC}}=0zIC=0---similar to previous results for the SW and Wolff algorithms.
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 zint,{\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}zint,​EW{\mathit{E}}^{\mathrm{W}}EW=\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 zint,{\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}zint,​ESW{\mathit{E}}^{\mathrm{SW}}ESW=\ensuremath{\beta}/\ensuremath{\nu} for the Ising model.