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Modern Physics Letters B

ON THE DETERMINATION OF CRITICAL POINT IN HIGHER-ORDER VARIATIONAL CUMULANT EXPANSION

Xumin Ou, Jing-Tian Ou, D. L. Lin

Published May 20, 1996• 7 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984996000584

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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.
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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.
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.
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.
A semi-phenomenological theory is developed on the basis of some assumptions in order to study the singularities of thermodynamic quantities in the vicinity of the Curie point. First, microscopic expressions are found for the coefficients in the expansion of the thermodynamic potential in terms of cumulants. Cumulants of all order are calculated in the one-dimensional Ising model, and the series expansion of the fourth order cumulant is also obtained in the two-dimensional Ising model. The three-dimensional Heisenberg model is also discussed, using the result of Rushbrooke et al. In all cases, the fourth order coefficient in the expansion of the thermodynamic potential tends to zero at the Curie points. And, in the last case, the eighth order coefficient diverges at the Curie point.