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Physical review. B, Condensed matter

Variational analyses of series expansions of the Potts model

H. H. Chen, Felix Lee, Yi-Chun Kao

Published Jul 1, 1995• 4 citations
Publisher

Abstract

A linked-cluster type of series expansion of the free energy with a variational parameter is developed to the eighth order for the q-state Potts model. The order parameter and the internal energy are then derived. By analyzing these series expansions, critical temperatures ${\mathit{T}}_{\mathit{c}}$, latent heats, and discontinuities of the order parameters at ${\mathit{T}}_{\mathit{c}}$ are determined for the cubic lattices.

DOI
10.1103/physrevb.52.39

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A linked-cluster type of series expansion of the free energy with a variational parameter is developed to the eighth order for the q-state Potts model. The order parameter and the internal energy are then derived. By analyzing these series expansions, critical temperatures Tc{\mathit{T}}_{\mathit{c}}Tc​, latent heats, and discontinuities of the order parameters at Tc{\mathit{T}}_{\mathit{c}}Tc​ are determined for the cubic lattices.
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.
The modified variational cumulant expansion (VCE) method valid for the whole temperature range and for arbitrary fields is developed to study the critical properties of the general q-state Potts model used to stimulate various magnetic systems. The series expansion is generated directly in terms of the Potts Hamiltonian. The free energy and the internal energy are calculated up to the third-order cumulant terms for the d-dimensional hypercubic q-state Potts model. The expression for the critical temperature Tc is derived in a closed form which allows to determine Tc up to arbitrary order cumulant terms. The modified VCE method eliminates the unphysical first-order phase transition at the mean-field critical point in the internal energy of the Potts model which plagues the conventional VCE method.
Abstract The thermodynamics of the q‐state Potts model is calculated by the cluster approximation. With two, three, and four particles as the basic cluster the free energy is obtained for any q. It is found that the transition is of the first order for all two‐dimensional and three‐dimensional Potts models. In the case of planar lattices for q = 3 and q = 4 it is found that the critical point approaches the Curie‐Weiss temperature but the discontinuity of the order parameter remains almost equal to that obtained by the mean‐field approximation.