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Journal of the Physical Society of Japan

Critical Exponents of Ising-Like Heisenberg Antiferromagnets on the Hexagonal Lattice

Seizo Watarai, Seiji Miyashita

Published Mar 15, 1998• 4 citations
Publisher

Abstract

Critical properties of the two successive phase transitions of Ising-like Heisenberg antiferromagnets on the hexagonal lattice are examined by the Monte Carlo method. In order to obtain the critical exponents, the finite size scaling methods are used for the lattice sizes of L =18 ∼ 48. Using the reweighting Monte Carlo method, the exponents at the higher critical temperature are estimated as 1/ν=1.481±0.019 and γ/ν=1.980±0.016, which agree with those of the 3DXY model. At the lower critical temperature, standard procedure of the reweighting method does not give values of the exponents with satisfactory precision. Thus we estimate the exponents carefully with aid of the ordinary Monte Carlo method at various temperatures near T c . The values of 1/ν≃ 1.48±0.006 and γ/ν≃1.97±0.050 are obtained and we conclude that those are consistent with the universality class of the 3DXY model.

DOI
10.1143/jpsj.67.816

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Critical properties of the two successive phase transitions of Ising-like Heisenberg antiferromagnets on the hexagonal lattice are examined by the Monte Carlo method. In order to obtain the critical exponents, the finite size scaling methods are used for the lattice sizes of L =18 ∼ 48. Using the reweighting Monte Carlo method, the exponents at the higher critical temperature are estimated as 1/ν=1.481±0.019 and γ/ν=1.980±0.016, which agree with those of the 3DXY model. At the lower critical temperature, standard procedure of the reweighting method does not give values of the exponents with satisfactory precision. Thus we estimate the exponents carefully with aid of the ordinary Monte Carlo method at various temperatures near T c . The values of 1/ν≃ 1.48±0.006 and γ/ν≃1.97±0.050 are obtained and we conclude that those are consistent with the universality class of the 3DXY model.
Phase transitions and critical properties in the antiferromagnetic Heisenberg model on a layered cubic lattice with allowance for intralayer next nearest neighbor interactions have been studied using the replica Monte Carlo algorithm. The character of phase transitions has been analyzed using the histogram method and the Binder cumulant method. It has been found that a transition from the collinear to paramagnetic phase in the model under study occurs as a second order phase transition. The statistical critical exponents of the specific heat α, susceptibility γ, order parameter β, and correlation radius ν, as well as the Fisher index η, have been calculated using the finite-size scaling theory. It has been shown that the three-dimensional Heisenberg model on the layered cubic lattice with allowance for the next nearest neighbor interaction belongs to the same universality class of the critical behavior as the antiferromagnetic Heisenberg model on a layered triangular lattice.
We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents \ensuremath{\nu},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\eta} from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to 1003{100}^{3}1003 spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
Abstract Computer simulation of the three-component Potts model on a hexagonal lattice was carried out using the Monte Carlo method. Systems with linear dimensions L × L = N, L = 20–320 in units of interatomic distances are considered. Based on the theory of finite-size scaling, the static critical exponents of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated. The data we obtained confirm that in the considered Potts model on a hexagonal lattice, a second-order phase transition is observed with critical exponents corresponding to the universality class of the three-component Potts model.
Computer simulation of the three-component Potts model on a hexagonal lattice was carried out using the Monte Carlo method. Systems with linear dimensions L × L = N, L = 10–320 in units of interatomic distances are considered. Based on the theory of finite-size scaling, the static critical exponents of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated. The data we obtained confirm that in the considered Potts model on a hexagonal lattice, a second-order phase transition is observed with critical exponents corresponding to the universality class of the three-component Potts model.
We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.