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Acta Physica Polonica A

Improved Phase Diagrams of Ashkin-Teller Model

P. Pawlicki, G. Kamieniarz, Piotr Kozłowski, Raf Dekeyser, Jos Rogiers

Published Aug 1, 1997• 1 citation
Publisher

Abstract

The phase diagram of the Askhii-Teller model in two dimensions is determined.Numerical calculations are performed for the simple square L x L lattice using transfer matrix technique.Exploiting finite-size scaling all unknown critical lines were obtained with good accuracy.An extended version of the Áskkin-Teller model is also considered within the molecular field renormalization group method and the critical surface for three-parameter odd-parity Hamiltonian is calculated.

DOI
10.12693/aphyspola.92.453

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The phase diagram of the Askhii-Teller model in two dimensions is determined.Numerical calculations are performed for the simple square L x L lattice using transfer matrix technique.Exploiting finite-size scaling all unknown critical lines were obtained with good accuracy.An extended version of the Áskkin-Teller model is also considered within the molecular field renormalization group method and the critical surface for three-parameter odd-parity Hamiltonian is calculated.
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The universal critical point ratio Q is exploited to determine positions of the critical Ising transition lines on the phase diagram of the Ashkin-Teller model on the square lattice. A leading-order expansion of the ratio Q in the presence of a nonvanishing thermal field is found from finite-size scaling and the corresponding expression is fitted by the accurate perturbative transfer-matrix data calculations for the L\ifmmode\times\else\texttimes\fi{}L square clusters with L\ensuremath{\leqslant}9.
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 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.