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Journal of Physics A Mathematical and General

Partial order in the self-dual Ashkin-Teller model

Katherine Seaton, Paul A. Pearce

Published Jul 7, 1989• 2 citations
Publisher

Abstract

Corner transfer matrices are used to calculate the order parameters of the self-dual Ashkin-Teller model on the square lattice. In the non-critical regime, it is found that the model is partially ordered, i.e. the magnetisations (s 1 ) and (t 1 ) vanish while the polarisation (s 1 t 1 ) is non-zero. The polarisation, which is given simply in terms of elliptic functions, exhibits an essential singularity at criticality.

DOI
10.1088/0305-4470/22/13/047

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Corner transfer matrices are used to calculate the order parameters of the self-dual Ashkin-Teller model on the square lattice. In the non-critical regime, it is found that the model is partially ordered, i.e. the magnetisations (s 1 ) and (t 1 ) vanish while the polarisation (s 1 t 1 ) is non-zero. The polarisation, which is given simply in terms of elliptic functions, exhibits an essential singularity at criticality.
The exact solution of an Ashkin-Teller model, with layer and spatial anisotropy and Z 2 *Z 2 symmetry, is presented. The model is related to the eight-vertex model by a sublattice or orbifold duality and it has the same free energy and magnetisation. The exact solution manifold is a three-dimensional submanifold of the full six-dimensional thermodynamic space. It admits a Z 3 symmetry relating three regimes each exhibiting different partial order. In the spatially isotropic case, these regimes are separated by three lines of continuously varying critical behaviour which coalesce at the four-state Potts critical point. The magnetisations and polarisation are derived using corner transfer matrices giving rise to the appearance of chiral and Virasoro characters. The associated critical exponents confirm the known values. In particular, one obtains the value x sigma =1/8 for the magnetic scaling dimension and identify the next-to-leading thermal field.
Abstract The critical behavior of a dimer model with an interaction favoring parallel dimers in each plaquette of the square lattice is studied numerically using the corner transfer matrix renormalization group algorithm. The critical exponents are known to depend on the chemical potential of vacancies, or monomers. At large average density of the latter, the phase transition becomes the first-order. We compute the scaling dimensions of both the order parameter and temperature in the second-order regime and compare them with the conjecture that the critical behavior is the same as the Ashkin–Teller model on its self-dual critical line.
In order to study the effect of the anisotropy parameter on the two coupled Ising models, we propose an Ashkin-Teller model (ATM) with spin variable S=1. Using an effective-field theory, we show that near the partially ordered phase PO1{\mathrm{PO}}_{1}PO1​ that appears in spin-1/2 ATM, this model presents a new partially ordered phase PO2{\mathrm{PO}}_{2}PO2​. The anisotropy parameter introduces a first-order phase transition between different phases. However, the model exhibits a rich variety of phase diagrams and higher-order critical points.
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.