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Journal of Physics A Mathematical and General

The three-dimensional XY model in P-fold random anisotropy

P Reed

Published Feb 7, 1991• 6 citations
Publisher

Abstract

The three-dimensional XY model in the presence of two- and threefold random anisotropy has been investigated by Monte Carlo simulation and finite-size scaling using system sizes 4 3 , 8 3 , 16 3 , 32 3 . The results are consistent with the existence of a conventional transition to a spin glass phase at T c =2.2 and with correlation exponent v=0.67. No evidence for quasiferromagnetism is found.

DOI
10.1088/0305-4470/24/3/004

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The three-dimensional XY model in the presence of two- and threefold random anisotropy has been investigated by Monte Carlo simulation and finite-size scaling using system sizes 4 3 , 8 3 , 16 3 , 32 3 . The results are consistent with the existence of a conventional transition to a spin glass phase at T c =2.2 and with correlation exponent v=0.67. No evidence for quasiferromagnetism is found.
The two-dimensional planar spin model in the presence of p-fold random anisotropy has been investigated using Monte Carlo simulation and finite size scaling. The system has been examined for p=2, 3, 4, 5, 6 and lattice sizes N=4 2 , 8 2 , 16 2 . The critical behaviour is found to depend on the value of p and is of two types. For p=3, 5, 6 the results indicate a transition to a quasiferromagnetic state at the same temperature as the pure system. At lower temperatures there is evidence for a second transition; however, the system remains quasiferromagnetic. For p=2, 4 the behaviour is different and results suggest a second-order transition to a spin glass phase and no quasiferromagnetism.
The three-dimensional anisotropic classical XY ferromagnet has been investigated by extensive Monte Carlo simulation using the Metropolis single spin flip algorithm. The magnetisation (M) and the susceptibility (χ) are measured and studied as functions of the temperature of the system. For constant anisotropy, the ferro-para phase transition has been found to take place at a higher temperature than that observed in the isotropic case. The system gets ordered at higher temperatures for higher values of the strength of anisotropy. The opposite scenario is observed in the case of random anisotropy. For all three different kinds of statistical distributions (uniform, Gaussian, and bimodal) of random anisotropy, the system gets ordered at lower temperatures for higher values of the width of the distribution of anisotropy. We have provided the phase boundaries in the case of random anisotropy. The critical exponents for the scaling laws M∼L−βν and χ∼Lγν are estimated through the finite size analysis.
We re-examine the spin-glass (SG) phase transition of the ± J Heisenberg models with and without random anisotropy D in three dimensions ( d = 3) using two complementary methods, i.e., (i) the defect energy method and (ii) the Monte Carlo method. We reveal that the conventional defect energy method is not convincing and propose a new method which considers the stiffness of the lattice itself. Using the method, we show that the stiffness exponent θ has a positive value (θ > 0) even when D = 0. Considering the stiffness at finite temperatures, we obtain the SG phase transition temperature of T SG ∼ 0.19 J for D = 0. On the other hand, a large scale MC simulation shows that, in contrast to the previous results, a scaling plot of the SG susceptibility χ SG for D = 0 is obtained using almost the same transition temperature of T SG ∼ 0.18 J . Hence we believe that the SG phase transition occurs in the Heisenberg SG model in d = 3.
We have studied the three-dimensional Ising spin glass with a $\ifmmode\pm\else\textpm\fi{}J$ distribution by Monte Carlo simulations. Using larger sizes and much better statistics than in earlier work, a finite-size scaling analysis shows quite strong evidence for a finite transition temperature Tc{T}_{c}Tc​ with ordering below Tc{T}_{c}Tc​. Our estimate of the transition temperature is rather lower than in earlier work, and the value of the correlation length exponent $\ensuremath{\nu}$ is somewhat higher. Because there may be (unknown) corrections to finite-size scaling, we do not completely rule out the possibility that Tc=0{T}_{c}=0Tc​=0 or that Tc{T}_{c}Tc​ is finite but with no order below Tc{T}_{c}Tc​. However, from our data, these possibilities seem less likely.