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Communications in Theoretical Physics

The Average Free Energy and the Effective Mean Field Hypothesis

Yongxin Song, Shao-Kang Zhou, pei-Ying Zhao

Published Jun 1, 1992• 5 citations
Publisher

Abstract

In this paper, an improvement in the variational-cumulant expansion method is made. For any model in lattice field theories, the average free energy can be calculated rigorously according to the generalized cumulant expansion, and the intensity of the effective external field can be determined by the effective mean field hypothesis. As a test, the Ising model is studied, and the results take a visible turn for the better.

DOI
10.1088/0253-6102/17/4/421

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Communications in Theoretical Physics•Jun 1, 1992•0 citations
In this paper, an improvement in the variational-cumulant expansion method is made. For any model in lattice field theories, the average free energy can be calculated rigorously according to the generalized cumulant expansion, and the intensity of the effective external field can be determined by the effective mean field hypothesis. As a test, the Ising model is studied, and the results take a visible turn for the better.
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.
We propose in this paper an effective mean-field theory, which makes the calculation for any observable being performed in a general form of cumulant expansion. The effective mean-field equation with reasonable physical annotation, which is obtained from the lower bound extreme value condition of free energy, can be applied to general statistic mechanics and Lagrangian quantum theory. By applying it to lattices, our theory, for various models containing nonuniform system and nonequilibrium phenomenon, can give a realistic scheme to study the nonperturbative effect order by order. As a testing example to two-dimensional Ising model, we obtain a result approaching the Onsager solution order by order.
It is shown for the Ising model that the linked-cluster expansion (LCE) is a special case of the variational cumulant expansion (VCE) in the high-temperature limit. The VCE was originally developed and formulated to treat lattice-gauge-field models. The free energy calculated from the VCE with a trial Hamiltonian of mean-field form in the high-temperature limit is in fact identical to that from the LCE. Thus, a more general expansion that is valid for the whole temperature range is established for the treatment of Ising systems.
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.