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Harbin Gongcheng Daxue Xuebao/Journal of Harbin Engineering University

Analytic study of phase Transition in q = 3,4 clock model

Yan-Cheng Zhao

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

Free energy ( W) and internal energy ( E ) up to the 4th order approximation, and specific heat ( C) and magnetization ( M ) up to the 3rd order approximation are calculated with the trial action of two vari-ational parameters, and the point of phase transition is obtained and the order of phase transition is discussed as well. The position of phase transition point given above agrees with the results of Monte Carlo ( MC)

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Free energy ( W) and internal energy ( E ) up to the 4th order approximation, and specific heat ( C) and magnetization ( M ) up to the 3rd order approximation are calculated with the trial action of two vari-ational parameters, and the point of phase transition is obtained and the order of phase transition is discussed as well. The position of phase transition point given above agrees with the results of Monte Carlo ( MC)
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 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.
We study the phase transition properties of the two-dimentional q-state clock model by an extensive Monte Carlo simulation. Several independent physical quantities, including the specific heat, the magnetization and the helicity modulus had been calculated for q > 5. The specific heat and the magnetization shows the behavior of two distinct phase transitions, including the higher phase transition, T2, between the disordered phase and the quasi-long-range ordered phase and the lower phase transition, T1, between the quasi-long-range ordered phase and the quasi-long-range ordered phase. The helicity modulus not only show that T2 is the Kosterlitz-Thouless phase transition but also has signature of phase transition around T1.
Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the phase transitions of the five-state clock model on the square lattice. The temperature dependence of the specific heat indicates the system has two phase transitions, as verified clearly by the correlation function at three representative temperatures. By calculating the magnetic susceptibility, we obtained only the upper critical temperature as T c2 = 0.9565(7). Investigating the fixed-point tensor, we precisely locate the transition temperatures at T c1 = 0.9029(1) and T c2 = 0.9520(1), consistent well with the Monte Carlo and the density matrix renormalization group results.
A Monte Carlo (MC) computer experiment for the analysis of first-order temperature-driven phase transitions in a system with one or many independently behaving order parameters is presented using the example of the three-dimensional (3D) Ashkin-Teller model, one of the important reference systems in statistical physics showing a rich and complex phase diagram. The properties of a number of quantities, such as magnetization, three types of cumulants, the internal energy, and its histogram, are exploited. The Lee and Kosterlitz concept proposed for strong first-order phase transitions in systems with one independent order parameter is significantly expanded to obtain results with comparable error bars in reasonable computation times at an arbitrary amount of latent heat. The proposed computer MC experiment uses parallel processing and both the Metropolis and recently formulated cluster algorithms. Arbitrarily weak to strong first-order phase transitions in the phase diagram region with ferromagnetic interactions are investigated and the latent heat associated with individual degrees of freedom is carefully computed. In the discussion of results, the behavior of our 3D system between that of the mean-field and that of the 2D one is bracketed and the role of the Potts point is clarified.