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Physical review. B, Condensed matter

Monte Carlo simulations of Ising-like phase transitions in the three-dimensional Ashkin-Teller model

Grzegorz Musiał, L. Dębski, G. Kamieniarz

Published Jun 28, 2002• 27 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.66.012407

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The phase transition line near the Ising point region is studied for the 3D standard AshkinTeller model on a cubic lattice.This model with a multicomponent order parameter is one of the important reference points in statistical physics since it shows an interesting and complicated phase diagram.The main motivation for our study was nonuniversal behavior suggested for this line.The large-scale Monte Carlo simulations using the Binder and Challa like cumulants are performed.Accurate analysis to exclude the latent heat inherence is applied.Specic behavior of the Challa like cumulants is discovered and its interpretation is proposed.The paper is closed with preliminary conclusions concerning the continuous but non-Ising character of these phase transitions in the lower part of the mixed phase region and the possiblity of the rst order on the line connecting it to the Ising point.

Acta Physica Polonica A•May 1, 2012•10 citations
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.
Monte Carlo simulations in the three-dimensional (3D) Ashkin-Teller model on a cubic lattice are performed in the regions of the two-parameter space diagram where Ising-type phase transitions are expected. The scaling behavior of the Binder cumulant Q and the magnetic susceptibility in the critical region are exploited. In simulations the periodic boundary conditions and the Metropolis algorithm are used. Starting from Ising critical exponents and applying the finite-size-scaling analysis of the cumulant Q with nonlinear corrections, the accurate positions of the critical couplings on the continuous phase transition lines are calculated. For these couplings the critical exponent y(h) is calculated analyzing the magnetic susceptibility in the framework of the finite-size scaling with corrections. The values of y(h) agree with the 3D Ising model value along two lines determined by the order parameter [ssigma] .
Abstract Large‐scale Monte Carlo simulations, based on the invariance of the Binder cumulant Q, for continuous phase transitions in the three‐dimensional Ashkin–Teller spin‐lattice model on a cubic lattice, have been performed. Using the universality hypotesis and the finite‐size‐scaling analysis, the Ising character of phase transitions from the antiferro‐ to paramagnetic phase, where the cumulant Q behavior is different than in the Ising model, is demonstrated. Some preliminary results demonstrating the existance of the tricritical points are also presented.
The complementary Monte Carlo and series expansions methods of computer simulations have been described to investigate the critical behaviour of the Ashkin-Teller model in three dimensions.In the first method the invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region has been exploited and some critical points on the phase boundaries have been calculated in the regions where the continuous transitions are expected.The continuity of the order parameter on the critical lines is verified by a finite size scaling analysis.Large-scale simulations have been performed on SGI Power Challenge XL and L supercomputers using the 64-bit random number generator.The numerically generated series expansions method is described for which the effective algorithm for generation of graphs starting from polygons and based on collapsing the unlinked vertices, is introduced.The new feature of our algorithm is that for each graph we introduce new links between unlinked vertices and we decorate bonds with new vertices, so that more complex graphs in an early stage of the graph generation procedure are obtained.The resulting set of graphs enables the application of the series expansions method and achievement of the precision of allocation of points on the phase diagram comparable to the precision of the Monte Carlo method, i.e. at least 3 decimal digits.
Our Monte Carlo (MC) method has been extended to determine the type of ordering, the character of phase transitions, and positions of the tricritical points in the three-dimensional Ashkin-Teller spin-lattice model on a cubic lattice, which is an example of a system with multicomponent order parameter. The precise and detailed analyses of the system's behavior at the ends of the lines of Ising-like phase transitions are performed and the corrected phase diagram in these regions is presented and discussed. The localizations of the tricritical points are also given. The specific behavior of the Binder cumulant when crossing a bifurcation point has been discovered and interpreted. Some general conclusions on methods of investigation of the critical behavior of a system with a multicomponent order parameter are closing the paper.
The phase transition line near the Ising point region is studied for the 3D standard AshkinTeller model on a cubic lattice.This model with a multicomponent order parameter is one of the important reference points in statistical physics since it shows an interesting and complicated phase diagram.The main motivation for our study was nonuniversal behavior suggested for this line.The large-scale Monte Carlo simulations using the Binder and Challa like cumulants are performed.Accurate analysis to exclude the latent heat inherence is applied.Specic behavior of the Challa like cumulants is discovered and its interpretation is proposed.The paper is closed with preliminary conclusions concerning the continuous but non-Ising character of these phase transitions in the lower part of the mixed phase region and the possiblity of the rst order on the line connecting it to the Ising point.