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arXiv:hep-lat/9310015 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Cluster method for the Ashkin-Teller model

Shai Wiseman, Eytan Domany

Published Nov 1, 1993• 55 citations
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Abstract

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 ${\mathit{z}}_{\mathrm{cluster}}$\ensuremath{\ge}\ensuremath{\alpha}/\ensuremath{\nu}, that was so far proved only for Potts models.

DOI
10.1103/physreve.48.4080

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Physical review. B, Condensed matter•Jun 28, 2002•27 citations
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.
The critical behavior of a bond-disordered Ashkin-Teller model on a square lattice is investigated by intensive Monte Carlo simulations. A duality transformation is used to locate a critical plane of the disordered model. This critical plane corresponds to the line of critical points of the pure model, along which critical exponents vary continuously. Along this line the scaling exponent corresponding to randomness \ensuremath{\varphi}=(\ensuremath{\alpha}/\ensuremath{\nu}) varies continuously and is positive so that the randomness is relevant, and different critical behavior is expected for the disordered model. We use a cluster algorithm for the Monte Carlo simulations based on the Wolff embedding idea, and perform a finite size scaling study of several critical models, extrapolating between the critical bond-disordered Ising and bond-disordered four-state Potts models. The critical behavior of the disordered model is compared with the critical behavior of an anisotropic Ashkin-Teller model, which is used as a reference pure model. We find no essential change in the order parameters' critical exponents with respect to those of the pure model. The divergence of the specific heat C is changed dramatically. Our results favor a logarithmic type divergence at Tc{\mathit{T}}_{\mathit{c}}Tc​, C\ensuremath{\sim}lnL for the random-bond Ashkin-Teller and four-state Potts models and C\ensuremath{\sim}ln lnL for the random-bond Ising model.
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}}.
We investigate the symmetric Ashkin-Teller (AT) model on the triangular lattice in the antiferromagnetic two-spin coupling region (J<0). In the J→-∞ limit, we map the AT model onto a fully packed loop-dimer model on the honeycomb lattice. On the basis of this exact transformation and the low-temperature expansion, we formulate a variant of worm-type algorithms for the AT model, which significantly suppress the critical slowing down. We analyze the Monte Carlo data by finite-size scaling, and locate a line of critical points of the Ising universality class in the region J<0 and K>0, with K the four-spin interaction. Further, we find that, in the J→-∞ limit, the critical line terminates at the decoupled point K=0. From the numerical results and the exact mapping, we conjecture that this "tricritical" point (J→-∞,K=0) is Berezinsky-Kosterlitz-Thouless-like and the logarithmic correction is absent. The dynamic critical exponent of the worm algorithm is estimated as z=0.28(1) near (J→-∞,K=0).
The results of the large-scale Monte Carlo simulations of the continuous phase transitions in the three-dimensional Ashkin-Teller model on a cubic lattice are presented in the two-parameter space. The invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region and the Metropolis algorithm are exploited. Using the universality hypothesis and the finite-size-scaling analysis, the accurate positions of the continuous phase-transition lines are calculated and their Ising character is confirmed. Approximate locations of the tricritical points are also estimated.