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Harbin Gongcheng Daxue Xuebao/Journal of Harbin Engineering University

Phase Transition of Six-State Potts Model

Sun Qiu

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

The phase transition of a 6_state vectorial Potts model has been studied by standard variational_cumulant expansion(VCE) method. With free energy(F) and internal energy (E) calculated to the fourth order approximation and specific heat (C) and magnetization (M) and susceptibility (χ) are calculated to the third order approximation with the trial action of two variational parameters. The position of the phase transition point establisbed is in good agreement with the results of Monte Carlo (MC). Which shows that VCE is convergent during the calculation of the 6_state vectorial potts model and the trial action is equivalent to the action of the system.

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The phase transition of a 6_state vectorial Potts model has been studied by standard variational_cumulant expansion(VCE) method. With free energy(F) and internal energy (E) calculated to the fourth order approximation and specific heat (C) and magnetization (M) and susceptibility (χ) are calculated to the third order approximation with the trial action of two variational parameters. The position of the phase transition point establisbed is in good agreement with the results of Monte Carlo (MC). Which shows that VCE is convergent during the calculation of the 6_state vectorial potts model and the trial action is equivalent to the action of the system.
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 modified variational cumulant expansion (VCE) method valid for the whole temperature range and for arbitrary fields is developed to study the critical properties of the general q-state Potts model used to stimulate various magnetic systems. The series expansion is generated directly in terms of the Potts Hamiltonian. The free energy and the internal energy are calculated up to the third-order cumulant terms for the d-dimensional hypercubic q-state Potts model. The expression for the critical temperature Tc is derived in a closed form which allows to determine Tc up to arbitrary order cumulant terms. The modified VCE method eliminates the unphysical first-order phase transition at the mean-field critical point in the internal energy of the Potts model which plagues the conventional VCE method.
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.