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Acta Physica Polonica A

Monte Carlo Simulations of the Ashkin-Teller Model

L. Dębski

Published May 1, 2000• 1 citation
Publisher

Abstract

Monte Carlo simulations are used to analyse the behaviour of the exteuded Ashkin-Teller model in two (2D) and three (3D) dimensions from the ratio of the square of the second moment of the order parameter to its fourth moment.The critical surface in three dimensional parameter space is calculated in regions where the Ising transitions are expected both for the ferro-and antiferromagnetic regions.

DOI
10.12693/aphyspola.97.859

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Monte Carlo simulations are used to analyse the behaviour of the exteuded Ashkin-Teller model in two (2D) and three (3D) dimensions from the ratio of the square of the second moment of the order parameter to its fourth moment.The critical surface in three dimensional parameter space is calculated in regions where the Ising transitions are expected both for the ferro-and antiferromagnetic regions.
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] .
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.
Monte Carlo simulations and renormalization-group ideas are used to analyze the behavior of the antiferromagnetic Potts model and the Ashkin-Teller model in three dimensions. A variety of continuous transitions, observed in the Monte Carlo data on the antiferromagnetic Potts model for simple cubic and body-centered-cubic lattices, are found to be consistent with $\ensuremath{\epsilon}$-expansion analyses of these models. A glassy "plastic crystal" phase is observed, along with the analog of the glass transition in the four-state antiferromagnetic Potts model. An $\ensuremath{\epsilon}$-expansion analysis of the Ashkin-Teller model is found to yield results consistent with those obtained by Ditzian et al. using series analysis and Monte Carlo simulations.
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.
A Monte Carlo (MC) computer experiment for the analysis of first-order temperature-driven phase transitions in a system with one or many independently behaving order parameters is presented using the example of the three-dimensional (3D) Ashkin-Teller model, one of the important reference systems in statistical physics showing a rich and complex phase diagram. The properties of a number of quantities, such as magnetization, three types of cumulants, the internal energy, and its histogram, are exploited. The Lee and Kosterlitz concept proposed for strong first-order phase transitions in systems with one independent order parameter is significantly expanded to obtain results with comparable error bars in reasonable computation times at an arbitrary amount of latent heat. The proposed computer MC experiment uses parallel processing and both the Metropolis and recently formulated cluster algorithms. Arbitrarily weak to strong first-order phase transitions in the phase diagram region with ferromagnetic interactions are investigated and the latent heat associated with individual degrees of freedom is carefully computed. In the discussion of results, the behavior of our 3D system between that of the mean-field and that of the 2D one is bracketed and the role of the Potts point is clarified.