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Journal of Physics Condensed Matter

A Monte Carlo study on the spin dynamics of the 2D+or-J Ising spin glass model

Koji Hukushima, K Nemoto

Published Mar 1, 1993• 10 citations
Publisher

Abstract

The authors examine the dynamical properties of the 2D+or-J Edwards-Anderson Ising spin glass in the anomalous phase which corresponds to the Griffiths phase in the diluted ferromagnetic system. By using Monte Carlo simulation they find that the autocorrelation function can be described as a function of a power of scaled time. This new scaling includes both the stretched exponential law for the intermediate stage and the logarithmic law for the early stage of relaxation. The relaxation time obtained from the scaling is found to diverge exponentially with exponent proportional to T -2 .

DOI
10.1088/0953-8984/5/9/023

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International Journal of Modern Physics C•Jun 1, 1996•1 citation
The authors examine the dynamical properties of the 2D+or-J Edwards-Anderson Ising spin glass in the anomalous phase which corresponds to the Griffiths phase in the diluted ferromagnetic system. By using Monte Carlo simulation they find that the autocorrelation function can be described as a function of a power of scaled time. This new scaling includes both the stretched exponential law for the intermediate stage and the logarithmic law for the early stage of relaxation. The relaxation time obtained from the scaling is found to diverge exponentially with exponent proportional to T -2 .
Slow dynamics in the ± J Ising model on a square lattice is studied by means of Monte Carlo simulation, in which the distribution function of (logarithms of) relaxation times, P ( x ) with x =ln τ , is obtained by analyzing individual spin auto-correlation functions one by one. At each temperature in the \lqGriffiths phase', namely in the range above the spin-glass transition temperature T c (=0) but below the Curie temperature of the corresponding pure ferromagnet Tpurec, P ( x ) exhibits a characteristic crossover behavior at around x c (=ln τ c ) which diverges toward T c . The contributions to the branch x ≤ x c are interpreted due to the ordinary critical slowing down process associated with T c (critical dynamics). The behavior of P ( x ) at x ≤ x c , on the other hand, is quite consistent with the picture of relaxation processes of large, rare clusters related to the expected Griffiths singularity at Tpurec (cluster dynamics).
We study the Monte Carlo simulation on the two-dimensional ± J Ising spin glass aiming to clarify the relation between behavior of individual spins and spin clusters in each sample and common spin glass properties of the system. Particularly relaxation times of individual spins are evaluated and their temperature dependences are examined. Making use of data obtained, we discuss in detail spin-freezing process in this model for spin glass. One of our most interesting findings is that spin relaxation processes at temperatures T≳0.8JT{\gtrsim}0.8JT≳0.8J obey Fulcher's law with T 0 ≃0.5 J , implying that the characteristic relaxation time of the system may diverge at T 0 .
A simple approximation procedure is introduced which quantitatively accounts for the low-temperature logarithmic decay in time of the autocorrelation function for the Edwards-Anderson model of an Ising spin glass. The relationship between the autocorrelation function and the susceptibility is used to derive the dependence on observation time of the susceptibility. The order parameter q(t) is calculated and is in good agreement with the Monte Carlo simulations on two- and three-dimensional systems with Gaussian bond distributions, which show that q(t) tends to zero for long times t.
Relaxation phenomena are studied for the ± J Ising spin glass in two dimensions by Monte Carlo method. For each spin, a representative relaxation time of the spin is estimated from the equilibrium autocorrelation function of the spin. Clusters of slowly relaxing spins are found to exist below the critical temperature of the nonrandom Ising ferromagnet. It is also found that the distribution of the relaxation times of spins is consistent with the prediction of the theory which assumes independent motion of clusters. This distribution explains the nonexponential decay of the averaged spin autocorrelation function as in the case of the diluted Ising ferromagnets.
Slow dynamics in the 'Griffiths phase' of the 3D ± J Ising spin glass is studied by means of Monte Carlo simulation, in which the spectra of (logarithms of) relaxation times, P (x) with x = ln τ, are analyzed in detail. At each temperature P (x) exhibits a characteristic crossover behavior at around a certain value xc which in turn tends to diverge toward the spin-glass critical temperature Tc. The contributions to the branch x ≲ xc are interpreted due to the ordinary critical slowing down process associated with Tc (critical dynamics). The behavior of P (x) at x ≲ xc, on the other hand, is consistent with the picture of relaxation processes of large, rare clusters which are related to the Griffiths singularity at the Curie temperature of the corresponding pure ferromagnet (cluster dynamics).