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Matematičeskie struktury i modelirovanie

Computer Simulation of the Nonequilibrium Critical Behavior of the Weakly Disordered Potts Model With the Number of Spin States Q=3

В. Н. Бородихин, M.M. Mikhailovskiy

Published Jan 1, 2019• 0 citations
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Abstract

In this paper, by the method of short-time dynamics the behavior of weakly disordered three-dimensional Potts model with the number of spin components equal to three was studied. The critical temperature value has been clarified. For spin concentrations 0.95 critical indexes z and ν were calculated.
DOI
10.24147/2222-8772.2019.3.45-53

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A computer simulation of the critical behavior of a two-dimensional highly diluted 3-state antiferromagnetic Potts model on a triangular lattice is performed. The calculations are done for systems with periodic boundary conditions at spin concentrations p equal to 0.70 and 0.65. Systems with linear dimensions L×L = N, L = 20–144 are considered. Based on the theory of finite-dimensional scaling, the static critical exponents of heat capacity α, susceptibility γ, order parameter β, and critical index ν for the correlation radius are calculated. It is numerically shown that the calculated critical exponents vary with changes in the spin concentration p, while the ratios β/ν and γ/ν remain unchanged within error, showing a weak universality of the critical behavior of disordered systems.
The results of computer simulations of the critical behavior of three-dimensional structures described by disordered Potts models with q = 3 and q = 4 spin states are discussed. Systems with linear dimensions L = 20–60 with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 are examined. It is found that in a three-dimensional Potts model with q = 3 spin states, introducing nonmagnetic impurities induces a second order phase transition. In a Potts model with q = 4, on the other hand, a shift from a first order phase transition to a second order phase transition is observed at high dilution (p = 0.65). Numerical values of the critical parameters are found using finite-size scaling theory and it is shown that they form a universality class corresponding to disordered systems.
The critical behavior of a two-dimensional weakly diluted antiferromagnetic Potts model with spin states q = 3 on a triangular lattice is computationally simulated. The systems with periodic boundary conditions were calculated, when spin concentrations p were 1.0 and 0.9. Systems with linear dimensions L × L = N and L = 9–144 were considered. Static critical exponents of heat capacity α, the susceptibility γ, the order parameter β, and critical exponent ν for a correlation radius are calculated based on a theory of finite-dimensional scaling.
The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality 2βν+γν=d\frac{{2\beta }}{\nu } + \frac{\gamma }{\nu } = dν2β​+νγ​=d . Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.
The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v .
The influence of quenched nonmagnetic impurities on phase transitions in the three-dimensional Potts model with the number of spin states q = 3 is investigated using the Wolff single-cluster algorithm of the Monte Carlo method. The systems with linear sizes L = 20–44 at the spin concentrations p = 1.0, 0.9, 0.8, and 0.7 are analyzed. It is demonstrated with the use of the method of fourth-order Binder cumulants that the second-order phase transition occurs in the model under consideration at the spin concentrations p = 0.9, 0.8, and 0.7 and that the first-order phase transition is observed in the pure model ( p = 1.0). The static critical exponents α (heat capacity), γ (susceptibility), β (magnetization), and ν (correlation length) are calculated in the framework of the finite-size scaling theory. The problem regarding the universality classes of the critical behavior of weakly diluted systems is discussed.