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arXiv:cond-mat/0604124 · Journal of Statistical Mechanics Theory and Experiment

Critical behaviour of the random-anisotropy model in the strong-anisotropy limit

Francesco Parisen Toldin, Andrea Pelissetto C, Ettore Vicari

Published Jun 5, 2006• 18 citations

Abstract

We investigate the nature of the critical behaviour of the random-anisotropy Heisenberg model (RAM), which describes magnetic systems with random uniaxial single-site anisotropy, such as some amorphous alloys of rare earths and transition metals. In particular, we consider the strong-anisotropy limit (SRAM), in which the Hamiltonian can be rewritten as the one of an Ising spin-glass model with correlated bond disorder. We perform Monte Carlo simulations of the SRAM on simple cubic lattices of linear size L , up to L = 30, measuring correlation functions of the replica–replica overlap, which is the order parameter at a glass transition. The corresponding results show critical behaviour and finite-size scaling. They provide evidence of a finite-temperature continuous transition with critical exponents η o = −0.24(4) and ν o = 2.4(6). These results are close to the corresponding estimates that have been obtained in the usual Ising spin-glass model with uncorrelated bond disorder, suggesting that the two models belong to the same universality class. We also determine the leading correction-to-scaling exponent, finding ω = 1.0(4).

DOI: 10.1088/1742-5468/2006/06/p06002

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We investigate the nature of the critical behaviour of the random-anisotropy Heisenberg model (RAM), which describes magnetic systems with random uniaxial single-site anisotropy, such as some amorphous alloys of rare earths and transition metals. In particular, we consider the strong-anisotropy limit (SRAM), in which the Hamiltonian can be rewritten as the one of an Ising spin-glass model with correlated bond disorder. We perform Monte Carlo simulations of the SRAM on simple cubic lattices of linear size L , up to L = 30, measuring correlation functions of the replica–replica overlap, which is the order parameter at a glass transition. The corresponding results show critical behaviour and finite-size scaling. They provide evidence of a finite-temperature continuous transition with critical exponents η o = −0.24(4) and ν o = 2.4(6). These results are close to the corresponding estimates that have been obtained in the usual Ising spin-glass model with uncorrelated bond disorder, suggesting that the two models belong to the same universality class. We also determine the leading correction-to-scaling exponent, finding ω = 1.0(4).

The critical behavior of the Random Anisotropy Axis model (RAM) is studied by the Monte Carlo simulation method in the limit of infinite anisotropy. Finite size scaling analysis of the data for equilibrium averages strongly suggests a zero temperature critical point in two dimensions and a finite temperature phase transition in three dimensions. The values of the critical exponents in both two and three dimensions indicate that RAM belongs to the same universality class as that of short-range Ising spin-glass models.

We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.

We perform high-statistics Monte Carlo simulations of three three-dimensional Ising spin glass models: the ± J Ising model for two values of the disorder parameter p , p = 1/2 and 0.7, and the bond-diluted ± J model for bond-occupation probability p b = 0.45. A finite-size scaling analysis of the quartic cumulants at the critical point shows conclusively that these models belong to the same universality class and allows us to estimate the scaling-correction exponent ω related to the leading irrelevant operator, ω = 1.0(1). We also determine the critical exponents ν and η. Taking into account the scaling corrections, we obtain ν = 2.53(8) and η = −0.384(9).

The random anisotropy axis model (RAM) in two dimensions (d=2) is studied in the limit of infinite anisotropy strength using the large-cell renormalisation group method. It is found that the model exhibits a 'zero-temperature phase transition', to a nonferromagnetic ground state, with critical exponents consistent with those of the d=2 Ising spin glass model. It is concluded that the RAM belongs to the universality class of the Ising spin glass, at least for d=2 and probably for all d<or=4.