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).
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 .
We have carried out Monte Carlo simulation on the 3-dimensional isotropic ± J spin glass model, by which the distribution function of relaxation times P (τ) is extracted from autocorrelation functions of all individual spins in the system. The obtained P (τ) gives clear evidence for the cluster dynamics due to the Griffiths singularity in a spin glass system as well as for the crossover between the cluster dynamics and the critical dynamics associated with the spin glass transition at T c (=0).
Non-exponential slow dynamics in the `Griffiths phase' of the three-dimensional (3D) ± J Ising spin-glass model is studied by means of Monte Carlo simulation. The distribution of relaxation times averaged over samples, \bar P ( x ) with x = lnτ, is calculated from spin auto-correlation functions simulated. It is found that \bar P ( x ) consists of two branches. The cross-over value between the branches, x c , diverges as T → T c , T c being the spin-glass transition temperature. Therefore τ c (= e x c ) is regarded as the critical relaxation time associated with the onset of the spin-glass order at T c , and the branch of \bar P ( x ) at x < x c is interpreted due to the ordinary critical slowing down process ( critical dynamics ). The other branch at x > x c causes the unusual slow dynamics peculiar to spin glasses. Various aspects of \bar P ( x ) of this branch examined are consistent with the following picture; compact regions with less frustrations than the system as a whole are thermally fluctuating with very small frequencies, and independently of each others ( cluster dynamics ).
The crossover between quasiequilibrium and nonequilibrium spin-glass dynamics has been studied in Monte Carlo simulations of two- and three-dimensional short-range Ising spin-glass systems. The spin system was quenched in zero field to a low temperature T. After equilibrating for a time
tw, a weak magnetic field was applied and the time dependence of the magnetization M(t) and the spin autocorrelation function q(t) was studied. Our results show that the relaxation rate S(t)=\ensuremath{\partial}M(t)/\ensuremath{\partial}
log10(t) exhibits a maximum at t\ensuremath{\approxeq}
tw. If the temperature is lowered or raised, immediately prior to the field application, the apparent age (
ta) of the system, as defined by the maximum in relaxation rate, is shifted to longer or shorter time scales, respectively. No significant differences between the aging behavior in two and three dimensions are found. The relation \ensuremath{\chi}(t)=[1-q(t)]/T holds only for short time scales (t\ensuremath{\ll}
tw), clear deviations are found at longer time scales (t>
tw). Since this relation should according to the fluctuation-dissipation theorem hold for a system in equilibrium, this result is interpreted as an indication of a crossover between equilibrium and nonequilibrium dynamics. Our results are qualitatively in agreement with experiments as well as with theoretical models for the nonequilibrium spin-glass dynamics.
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.