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Phase Transitions

Investigation of critical phenomena of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions by the Monte Carlo method

Рамазанов Магомедшейх Курбанович, Akai Kurbanovich Murtazaev

Published Aug 16, 2017• 5 citations
Publisher

Abstract

Phase transitions and critical properties of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions are investigated by the replica Monte Carlo method. Estimations are made for the magnitude relation of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J2/J1 in the value ranges of r ∈ [0.0, 1.0]. The phase diagram of the dependence of critical temperature on the next-nearest-neighbor interaction has been plotted. The static critical exponents of the heat capacity, the susceptibility, the ordering parameter and the correlation length as well as the Fisher exponent are calculated by means of the finite-size scaling theory. The universality class of the critical behavior of this model is revealed to remain within the limits of values r ∈ [0.0, 0.4]. It is found that the change in the next-nearest-neighbor interaction value in the range of r ∈ [0.9, 1.0] leads to another critical behavior.

DOI
10.1080/01411594.2017.1359836

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Phase transitions and critical properties of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions are investigated by the replica Monte Carlo method. Estimations are made for the magnitude relation of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J2/J1 in the value ranges of r ∈ [0.0, 1.0]. The phase diagram of the dependence of critical temperature on the next-nearest-neighbor interaction has been plotted. The static critical exponents of the heat capacity, the susceptibility, the ordering parameter and the correlation length as well as the Fisher exponent are calculated by means of the finite-size scaling theory. The universality class of the critical behavior of this model is revealed to remain within the limits of values r ∈ [0.0, 0.4]. It is found that the change in the next-nearest-neighbor interaction value in the range of r ∈ [0.9, 1.0] leads to another critical behavior.
Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.
The critical properties of the antiferromagnetic layered Ising model on a cubic lattice with regard to the nearest-neighbor and next-nearest-neighbor interactions are investigated by the Monte Carlo method using the replica algorithm. The investigations are carried out for the ratios of exchange nearest-neighbor and next-nearest-neighbor interactions r = J 2/J 1 in the range of 0 ≤ r ≤ 1.0. Using the finite-size scaling theory, the static critical indices of specific heat α, order parameter β, susceptibility γ, correlation radius ν, and Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is retained in the range of 0 ≤ r ≤ 0.4. It is established that the change in the next-nearest-neighbor interaction value in this model in the range of r > 0.8 leads to the same universality class as the three-dimensional fully frustrated Ising model on the cubic lattice.
The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.
The replica Monte Carlo method has been used to investigate the critical behavior of a threedimensional antiferromagnetic Ising model on a body-centered cubic lattice, taking into account interactions of the adjacent behind neighbors. Investigations are carried out for the ratios of the values of exchange interactions behind the nearest and next nearest neighbors k = J 2/J 1 in the range of k ∈ [0.0, 1.0] with the step Δk = 0.1. In the framework of the theory of finite-dimensional scaling the static critical indices of heat capacity α, susceptibility γ, of the order parameter β, correlation radius ν, and also the Fisher index η are calculated. It is shown that the universality class of the critical behavior of this model is kept in the interval of k ∈ [0.0, 0.6]. It is established that a nonuniversal critical behavior is observed in the range k ∈ [0.8, 1.0].
The phase transitions in antiferromagnetic Ising model are studied on a body-centered cubic lattice by taking the interactions of next-nearest neighbors into account. The model is investigated on basis of the replica Monte Carlo algorithm and the histogrammic analysis of data. The diagram of the critical temperature dependence on an interaction value of next-nearest neighbors is plotted. The studied model reveals the phase transitions of second order. A static magnetic critical indices is calculated using the finite-dimensional scaling theory.