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Journal of Applied Physics

Study of critical properties of the Potts model by the modified variational cumulant expansion method

Nail G. Fazleev, Hao Che, John L. Fry, D. L. Lin

Published Nov 15, 1994• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.358262

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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.
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The variational-cumulant-expansion (VCE) method is adopted to calculate the Heisenberg model. The free energy of the model is expanded to the third order, and the analytical formulas for the critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ are given for each order. The first order of VCE corresponds to the mean-field theory, the second order gives the same result for Tc{\mathit{T}}_{\mathit{c}}Tc​ as the 1/z expansion, and the third order gives a more complicated correction. As a comparison, the results of the two-dimensional (2D) Ising model for VCE to the seventh order are shown. We carry out a trial to predict the critical temperature of the infinite order of the VCE, and get a very accurate result for the 2D Ising model. A prediction of Tc{\mathit{T}}_{\mathit{c}}Tc​ for the Heisenberg model is also made.
The variational-cumulant expansion (VCE) method is used to calculate the quantum XY model, for which the trial action does not commute with the action , with arbitrary magnitude of spin on various one-, two- and three-dimensional lattices. The free energy of the model is expanded to the third order and the analytical formulae of critical temperature are given for each order. To testify this method in a wider range, the Heisenberg model on different lattices is also calculated, which has more comparable results. Weighted fitting method of statistical mathematics is adopted to predict of infinite order.
An effective mean-field theory based on cumulant expansion is expanded to deal with the first-order transition of the Potts model on general d-dimensional hypercubic lattice in this paper. It is a generalized mean-field method developed from the cumulant expansion of average free energy. As we shall see, it gives the traditional mean-field result at its first-order approximation and the effective mean field introduced by this method will be amended with the approximation's order increasing. For a full view of this method, we have carried out calculations up to the seventh-order approximation on a square (d=2)(d=2)(d=2) lattice since some known exact quantites of this case are terminals that should be approached. The results show this method has the potential to be a useful tool to calculate thermodynamical observables of spin systems.
To study the variational-cumulant expansion (VCE) method, the 2-dimensional Ising model has been selected. The model is evaluated to the sixth order approximation with a scheme for the enumeration of connected diagrams. The internal energy , specific heat and spontaneous magnetization have been calculated. A comparison with the exact results shows that thermodynamic and magnetic quantities converge to the exact values rather quickly and the position of phase transition point approaches to the exact pointstand order by order.