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Communications in Theoretical Physics

Variational-Cumulant Expansion up to the Sixth Order Approximation on 2-Dimensional Ising Model

Ruijie Liu, T. L. Chen, Wu-Qun Huang

Published Oct 1, 1993• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/20/3/283

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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.
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.
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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.
The phase transition of the 5-state clock model has been studied by standard variational-cumulant expansion (VCE) method. We calculated the free energy and the internal energy to the fourth order approximation and the specific heat to the third order approximation with the trial action of one variational parameter. The position of the phase transition point given above is in agreement with the results of the Monte Carlo (MC). We also calculated the model to the third order approximation with the trial action of two variational parameters. The comparison of the results for one variational parameter with that for two is given. From this, we can see how the choice of the trial action affects the result and the trial action must be equivalent to the action of the system. All above has shown that the VCE is convergent in the calculation of the 5-state clock model.
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.