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Communications in Theoretical Physics

Analysis and Discussion About How to Determine the Variational Parameters in the Variational-Cumulant Expansion

Ruijie Liu, Chen Tian-Lun

Published Jan 30, 1995• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/23/1/37

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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.
In order to investigate the variational-cumulant expansion (VCE) method, the 2-dimensional Ising model is expanded to the seventh order by adopting the method of enumerating connected diagrams, in whose process, how to determine the variational parameter is analyzed. The internal energy , specific heat and spontaneous magnetization are calculated from the first order to the seventh order. The results show that the thermodynamic and magnetic quantities converge to the exact values very quickly and the critical point approaches the exact one order by order. The curve-pairing phenomenon is watched between neighboring order curves at high temperature region.
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.
The standard variational-cumulant expansion (VC) combined with a scheme for the enumeration of connected diagrams is employed to study the simple Ising model with the second-order phase transition. This model is evaluated to the 5th order approximations. A comparison with the 2-dimensional exact results shows that thermodynamic quantities converge rather slowly to the exact values in the intermediate region. A strict analytical calculation of the critical temperature with the 5th order approximation both for and cumulant expansion suggests that VC can achieve rather high standard accuracy if the variational parameters could be determined up to higher degrees of corrections rather than the first one. Anomalies are found for and at extremely low temperature. A detailed study of implies that thermodynamic formulae should be carefully chosen in VC approach so as to avoid these anomalies.
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.
With the variational-cumulant expansion (VCE) method, the critical temperature and spontaneous magnetization of the spatially anisotropic XY model are calculated up to the fourth order. The variational parameter is determined by the complete variational method and the results are compared with both the Monte Carlo results and the self-consistent harmonic approximation (SCHA) results. It is shown that the VCE method can give a better result than the SCHA method when the coupling between adjacent layers is small.
To the lowest order approximation, the method of variational cumulant expansion has been successful in the determination of the critical point of a spin system on the Ising model. In the second order calculation, unphysical phase transition arises. In this letter, we first analyze the origin of unphysical phase transitions in high-order variational cumulant expansion calculations. We then explain the basis on which a conjecture is proposed that to any order m of the variational cumulant expansion, the critical point is given by the bifurcation point of the free energy. The theory is finally verified numerically for the two-dimensional case.