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Computational Methods in Science and Technology

METHODS OF COMPUTER SIMULATIONS OF PHASE TRANSITIONS IN THE ASHKIN-TELLER MODEL

Grzegorz Musiał, L. Dębski, G. Kamieniarz, Jos Rogiers

Published Jan 1, 2001• 0 citations
Publisher

Abstract

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.

DOI
10.12921/cmst.2001.07.01.67-82

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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.
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] .
High- and low-temperature series, Monte Carlo simulations, mean-field calculations, and renormalization-group ideas are used to analyze the behavior of the Ashkin-Teller model in three dimensions. The data generated by series and Monte Carlo methods in the neighborhood of the decoupling point are consistent with the renormalization-group conclusion that this system cannot show a line of continuously varying exponents. Thus the phase diagram for the d=3d=3d=3 system is quite different from that in d=2d=2d=2. Indeed a new phase (not seen in d=2d=2d=2) is observed, one in which the system spontaneously breaks its natural symmetry between its two kinds of Ising spins. A variety of continuous transitions are observed and explained in terms of an analysis which is fully consonant with renormalization-group concepts.
Two nonperturbative methods by means of the transfer-matrix finite-size-scaling and Monte Carlo simulations are used to investigate the spin-1 Ashkin-Teller model (ATM). We have obtained rich phase diagrams with first- and second-order phase transitions with several multicritical points of higher order. Also this model exhibits a partially ordered phase PO2{\mathrm{PO}}_{2}PO2​ which does not exist in the spin-1/2 Ashkin-Teller model (ATM). Finally, the critical behavior of this model is discussed.
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.