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Journal of Physics A Mathematical and General

Universality in a generalised Potts model

Ian Graham Enting

Published Nov 1, 1980• 12 citations
Publisher

Abstract

A three-state Potts model with pure three-site interactions is analysed using low-temperature series expansions. Universality predicts that the exponents alpha ' and beta should be the same as for the 'hard-hexagons' lattice gas: 1/3 and 1/9 respectively. The series estimates are consistent with these predictions, and the estimates of gamma ' are consistent with the scaling value of 14/9.

DOI
10.1088/0305-4470/13/11/006

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Физика металлов и металловедение•Jul 1, 2023•0 citations
A three-state Potts model with pure three-site interactions is analysed using low-temperature series expansions. Universality predicts that the exponents alpha ' and beta should be the same as for the 'hard-hexagons' lattice gas: 1/3 and 1/9 respectively. The series estimates are consistent with these predictions, and the estimates of gamma ' are consistent with the scaling value of 14/9.
Thirteen terms are presented of the low-temperature series for the free energy of the three-state Potts model in arbitrary external fields. Extrapolation of the series for specific heat, order parameter, and susceptibilities indicates that the transition in zero field is continuous (in contrast to the Landau prediction) with exponents alpha = alpha '=0.05+or-0.10, beta =0.10+or-0.01, gamma ' approximately=1.5+or-0.2 and gamma ' perpendicular to =1.1+or-0.1. These conclusions suggest the presence of an 'anomalous' tricritical point in the model; the associated phase diagram is discussed and contrasted with phenomenological predictions.
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.
Computer simulation of the three-component Potts model on a hexagonal lattice was carried out using the Monte Carlo method. Systems with linear dimensions L × L = N, L = 10–320 in units of interatomic distances are considered. Based on the theory of finite-size scaling, the static critical exponents of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated. The data we obtained confirm that in the considered Potts model on a hexagonal lattice, a second-order phase transition is observed with critical exponents corresponding to the universality class of the three-component Potts model.