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Journal of Physics A Mathematical and General

Series analysis of the 3-state Potts model in (2+1) dimensions

Chris J. Hamer, J. Oitmaa, Z Weihong

Published Apr 7, 1992• 11 citations
Publisher

Abstract

Both high-temperature and low-temperature series are used to locate and characterize the first-order transition in the 3-state Potts model in (2+1) dimensions on both square and triangular lattices. Estimates are presented for the vacuum energy, latent heat, magnetization, susceptibility and mass gap at the transition. The spontaneous magnetization and latent heat appear to display an 'approximate universality' at this weak first-order transition.

DOI
10.1088/0305-4470/25/7/023

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Series studies of the four-state Potts model

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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.

Journal of Physics A Mathematical and General•Sep 1, 1979•24 citations
Both high-temperature and low-temperature series are used to locate and characterize the first-order transition in the 3-state Potts model in (2+1) dimensions on both square and triangular lattices. Estimates are presented for the vacuum energy, latent heat, magnetization, susceptibility and mass gap at the transition. The spontaneous magnetization and latent heat appear to display an 'approximate universality' at this weak first-order transition.
Both 'high-' and 'low-temperature' perturbation series are used to locate and characterize the first-order transition in the three-state Potts model in (3+1) dimensions on the simple-cubic, body-centred cubic and face-centred cubic lattices. Estimates are presented for the vacuum energy, 'latent heat', magnetization, susceptibility and mass gap at the transition. The results show a remarkable degree of universality between the different lattices, and a 'law of corresponding states' is followed very closely.
For paper II, see ibid, vol. 27, p.1503 (1994). The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the 3-state Potts model on the simple cubic lattice to order z 43 and the high-temperature expansion of the partition function to order v 21 . We use the numerical data to show that the transition is first-order, and estimate the latent heat, the discontinuity in the magnetization, and a number of other critical parameters.
Monte Carlo simulations are performed for first-order phase-transition models. The three-dimensional three-state Potts model has a weak first-order transition. For this model we calculate the density of states on L3{\mathit{L}}^{3}L3 block lattices (L up to size 36) and obtain high-precision estimates for the leading partition-function zeros. The finite-size-scaling analysis of the first zero exhibits the expected convergence of the critical exponent \ensuremath{\nu} toward 1/D for large L; in particular, we find \ensuremath{\nu}=2.955(26) from our two largest lattices. Analysis of our specific-heat Cv{\mathit{C}}_{\mathit{v}}Cv​ data yields l=0.080 31(26) for the latent heat. Along another line of approach, we calculate the mass gap m=1/\ensuremath{\xi} (\ensuremath{\xi} is the correlation length) for cylindrical L2{\mathit{L}}^{2}L2Lz{\mathit{L}}_{\mathit{z}}Lz​ lattices (L up to 24 and Lz{\mathit{L}}_{\mathit{z}}Lz​=256). The finite size-scaling analysis of these results is also consistent with the convergence of \ensuremath{\nu} toward 1/D, but that the limiting value is 1/D is not yet conclusively established. Some theoretical arguments favor \ensuremath{\nu}\ensuremath{\rightarrow}0 in case of a first-order transition in a cylindrical ${\mathit{L}}^{\mathit{D}\mathrm{\ensuremath{-}}1}$\ensuremath{\infty} geometry. Therefore, we also applied our approach to the 2D ten-state Potts model, which is known to have a strong first-order transition. In this case we find unambiguous evidence in favor of 1/D as the limiting value.
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.