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Journal of the Physical Society of Japan

Critical Dynamics and Cluster Dynamics of the 3D ±JIsing Spin-Glass Model in Its `Griffiths Phase'

達雄 小森, Hajime Takayama

Published May 15, 1997• 2 citations
Publisher

Abstract

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 ).

DOI
10.1143/jpsj.66.1472

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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 ).
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).
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).
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).
First results of massive Monte Carlo simulations of the d=3 Ising spin glass with ± J bond distribution, performed on a fast special purpose computer, are discussed. Equilibrium static and dynamic correlation functions have been recorded for temperatures T/J=1.10 to 2.00. A qualitative change in the behavior of the system, and the best fits for the spin glass correlation length and relaxation time favor a continuous phase transition at Tc/J≊1.20±0.05. We find no evidence for the Tc=0 freezing picture.
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 .