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Journal of Physics A Mathematical and General

Approximate universality at a first-order transition-the three-state Potts model in (3+1) dimensions

Zheng Weihong, Chris J. Hamer, J. Oitmaa

Published Sep 7, 1994• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/27/17/013

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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.
Both weak-coupling and strong-coupling series expansions are calculated for the Ising model in a transverse field at zero temperature in three dimensions. Series are obtained for the ground-state energy, the magnetization, the susceptibility, and the energy gaps, on the simple cubic, the body-centred cubic and the face-centred cubic lattices. The analysis of the critical behaviour is consistent with the behaviour predicted by renormalization group theory for the four-dimensional simple Ising model. There is a remarkable degree of universality between all three lattices.
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.