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Communications in Theoretical Physics

Variational-Cumulant Expansion on the Five-State Clock Model

Wen-Jun Ma, Chen Tian-Lun

Published Sep 1, 1993• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0253-6102/20/2/171

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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.
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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 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.