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Communications in Theoretical Physics

A New Principle to Determine the Variational Parameter for the Variational–Cumulant Expansion

Qing Wang, Chen Tian-Lun

Published Jan 30, 1999• 3 citations
Publisher

Abstract

The key point of the variational-cumulant expansion is the determination of the variational parameter. In this paper, we present the improved mean-field hypothesis (IMFH) which is the base to determine the parameter. The new method derived from the IMFH shows advantage over previous methods. The critical temperature and some thermal dynamical functions for the Heisenberg model are calculated with the new method.

DOI
10.1088/0253-6102/31/1/79

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The key point of the variational-cumulant expansion is the determination of the variational parameter. In this paper, we present the improved mean-field hypothesis (IMFH) which is the base to determine the parameter. The new method derived from the IMFH shows advantage over previous methods. The critical temperature and some thermal dynamical functions for the Heisenberg model are calculated with the new method.
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.
Four methods of determining the variational parameter in the variational-cumulant expansion (VCE) are studied on the 2-dimensional Ising model. A comparison with the exact result shows that the main-variation method is the best one at low temperature, the complete-variational method is better than others at high temperature, the accumulation-point method is the best one to approximate the intermediate coupling region, and each of the methods is analyzed from the original formulas or conjectures. Some suggestions are given in order to get optimal results. The VCE is also analyzed from a point of view of symmetry breaking for the first time.
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.