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Journal of Physics A Mathematical Nuclear and General

Three-state Potts model and anomalous tricritical points

Joseph P. Straley, Michael Ellis Fisher

Published Sep 1, 1973• 190 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/6/9/007

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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.
The phase diagram of the two- and three-state Potts model with infinite-range interactions in the external field is analyzed by studying the partition function zeros in the complex field plane. The tricritical point of the three-state model is observed as the approach of the zeros to the real axis at the nonzero field value. Different regimes, involving several first- and second-order transitions of the complicated phase diagram of the three-state model, are identified from the scaling properties of the zeros closest to the real axis. The critical exponents related to the tricritical point and the Yang-Lee edge singularity are well reproduced. Calculations are extended to the negative fields, where the exact implicit expression for the transition line is derived.
The four-state Potts model is studied via series expansions. Free energy and susceptibility high-temperature series are calculated for hypercubic lattices of general dimensionality. Estimates of critical temperatures and critical indices gamma and alpha are obtained as functions of dimensionality. The free energy, magnetisation, and susceptibility low-temperature series are calculated for d=4. The high-and low-temperature series together indicate that the phase transition in this dimensionality is first order. From this result the authors suggest that the apparent second-order transitions shown by the susceptibility for d>or=4 might be explained as spinodal pseudotransitions.
The problem of the type of the phase transition in the three-dimensional weakly diluted Potts model with the number of spin states q = 3 has been investigated by the Monte Carlo method. The temperature dependences of the Binder cumulants, energy, magnetization, specific heat, and susceptibility have been calculated. It is found that the second-order phase transition occurs in a system at the spin concentration p = 0.9. The critical exponents of the magnetization (β), specific heat (α), and susceptibility (γ) and the critical correlation-length exponent v were calculated on the basis of the finite-size scaling theory at p = 0.9.