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Journal of Physics A Mathematical and General

A renormalisation group investigation of the Ashkin-Teller-Potts model

S E Ashley

Published Oct 1, 1978• 11 citations
Publisher

Abstract

The Ashkin-Teller-Potts model is examined using two renormalisation group methods: (a) the cluster expansion; and (b) the lower bound variational method. The critical surface is found to be in agreement with that conjectured by Wu and Lin (1974). The authors find a very strong indication of a line of fixed points along which they obtain a continuous variation of the specific heat exponent with the four-spin coupling. A comparison is made with the eight-vertex model.

DOI
10.1088/0305-4470/11/10/019

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The Ashkin-Teller-Potts model is examined using two renormalisation group methods: (a) the cluster expansion; and (b) the lower bound variational method. The critical surface is found to be in agreement with that conjectured by Wu and Lin (1974). The authors find a very strong indication of a line of fixed points along which they obtain a continuous variation of the specific heat exponent with the four-spin coupling. A comparison is made with the eight-vertex model.
The (1+1)-dimensional Ashkin-Teller model is studied by the renormalisation group method and by the finite lattice extrapolation technique. The possible phases and their boundaries in an extended space of couplings are determined from the degeneracy structure of the low-lying levels in the thermodynamic limit. The critical behaviour is investigated for ferromagnetic couplings using increasing block sizes in the renormalisation transformation. The line of fixed points and its bifurcation at the Potts fixed point is studied.
A cluster Monte Carlo algorithm for the Ashkin-Teller (AT) model is constructed according to the guidelines of a general scheme for such algorithms. Its dynamical behavior is tested for the square-lattice AT model. We perform simulations on the line of critical points along which the exponents vary continuously, and find that critical slowing down is significantly reduced. We find continuous variation of the dynamical exponent z along the line, following the variation of the ratio of specific-heat and correlation-length exponents \ensuremath{\alpha}/\ensuremath{\nu}, in a manner which satisfies the Li-Sokal bound zcluster{\mathit{z}}_{\mathrm{cluster}}zcluster​\ensuremath{\ge}\ensuremath{\alpha}/\ensuremath{\nu}, that was so far proved only for Potts models.
A renormalisation group approach developed for the percolation problem by Young and Stinchcombe (1975) is extended to yield magnetic indices. A real-space cluster calculation is carried out for the Ashkin-Teller-Potts model (Ashkin & Teller 1964, Potts 1952) with general S, allowing contact with results for the Ising model (S to 2) as well as with the previously published thermal indices of the percolation problem (S to 1). For the percolation problem, the magnetic index obtained is 2.06, about 10% larger than current series estimates, and results in gamma =2.87.
The critical behavior of the sss-state Ashkin-Teller-Potts model on a square lattice is studied for s=3,4,and5s=3, 4, \mathrm{and} 5s=3,4,and5, using a variational renormalization-group transformation of the type introduced by Kadanoff. For the s=3s=3s=3 and s=4s=4s=4 models, the transformation correctly predicts a continuous phase transition. The calculated values of the critical exponents and the transition temperature are consistent with the results obtained from other approaches. For s=5s=5s=5, however, the transformation fails to detect the first-order transition that is known to occur.
Abstract We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a q-state Potts model coupled to an r-state Potts model in two dimensions. For integer values of q and r the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin–Teller case ( q = r = 2 ). Since our results extend to continuous values of the number of states, we can access the limit r → 1 corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot in general be obtained from those of Fortuin–Kasteleyn clusters by analytical continuation.