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Communications in Theoretical Physics

Analytic Study of XY-Ising Model

Chen HuaSheng, Zhu Yunlun

Published Jul 30, 2000• 0 citations
Publisher

Abstract

We apply the variational cumulant expansion to study the phase transition of a two-dimensional coupled XY-king model. This model can describe several different systems with underlying continuous and Z 2 symmetries. We calculate the free energy, internal energy and specific heat to the third-order approximation with the trial action of two variational parameters. A comparison with Monte Carlo data is also presented.

DOI
10.1088/0253-6102/34/1/179

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We apply the variational cumulant expansion to study the phase transition of a two-dimensional coupled XY-king model. This model can describe several different systems with underlying continuous and Z 2 symmetries. We calculate the free energy, internal energy and specific heat to the third-order approximation with the trial action of two variational parameters. A comparison with Monte Carlo data is also presented.
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 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 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.
We study, by a Migdal-Kadanoff approximation and Monte Carlo simulations, the phase diagram of a two-dimensional coupled XY-Ising model. This model can describe phase transitions in different systems with underlying continuous and Z2{\mathit{Z}}_{2}Z2​ symmetries. Depending on the parameters, we find separate XY, Ising and first-order transitions. Also, a line of continuous transitions is found with simultaneous loss of XY and Ising order and varying critical exponents. The fully frustrated XY and Josephson-junction systems can be considered to lie along different paths in the model which can result in nonuniversal behavior if the transition is a single one.
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.