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Journal of Physics A Mathematical and General

Upper and lower bounds on dynamic correlations in the Griffiths phase

Alan J. Bray

Published Feb 7, 1989• 27 citations
Publisher

Abstract

Dilute ferromagnetic spin systems are considered in the Griffiths phase below the percolation threshold. It is shown that, for relaxational dynamics, the spin autocorrelation functions are bounded both above and below by functions of the form exp(-A(1n t) d (d-1)/) (Ising systems) and exp(-Bt 1 2/) (Heisenberg systems), establishing these as the correct asymptotic forms.

DOI
10.1088/0305-4470/22/3/004

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Journal of the Physical Society of Japan•Aug 15, 1995•2 citations
Dilute ferromagnetic spin systems are considered in the Griffiths phase below the percolation threshold. It is shown that, for relaxational dynamics, the spin autocorrelation functions are bounded both above and below by functions of the form exp(-A(1n t) d (d-1)/) (Ising systems) and exp(-Bt 1 2/) (Heisenberg systems), establishing these as the correct asymptotic forms.
Nonperturbative contributions are shown to produce anomalously slow relaxation in dilute ferromagnets (or antiferromagnets), both in the paramagnetic and ``Griffiths'' phases. In the paramagnetic phase, the pseudoscaling form C(t)\ensuremath{\sim}exp{-td/(d+z){\mathit{t}}^{\mathit{d}/(\mathit{d}+\mathit{z})}td/(d+z)f(t/${\ensuremath{\xi}}_{\mathit{p}}^{\mathit{d}+\mathit{z}}$)} is predicted for the spin autocorrelation function, with ${\ensuremath{\xi}}_{\mathit{p}}$ the correlation length of the nondilute system. In the Griffiths phase, C(t)\ensuremath{\sim}exp{-A1nt${)}^{\mathit{d}/(\mathit{d}\mathrm{\ensuremath{-}}1)}$} and exp{-(Bt)1/2{)}^{1/2})1/2} are predicted for Ising Heisenberg systems, respectively, for t\ensuremath{\gg}${\ensuremath{\xi}}_{\mathit{p}}^{\mathit{d}+\mathit{z}}$.
The dynamics of a random Ising ferromagnet is considered, for a general distribution P(J)P(J)P(J) of exchange interactions, in the Griffiths phase. The autocorrelation function C(t)C(t)C(t) is found to have the previously derived asymptotic form, $C(t)\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}A{(\mathrm{ln}t)}^{\frac{d}{(d\ensuremath{-}1)}}]$, where ddd is the spatial dimension, only if $\ensuremath{-}\mathrm{ln}[P(J)]$ grows faster than ${J}^{\frac{d}{(d\ensuremath{-}1)}}$ as $J\ensuremath{\rightarrow}\ensuremath{\infty}$. Otherwise, the asymptotic form of C(t)C(t)C(t) depends explicitly on the form of P(J)P(J)P(J) at large JJJ.
Arguments are given that, for random spin systems, the density of states \ensuremath{\rho}(\ensuremath{\mu}) of the inverse of the susceptibility matrix vanishes as \ensuremath{\rho}(\ensuremath{\mu})\ensuremath{\sim}exp(-A/\ensuremath{\mu}), for \ensuremath{\mu}\ensuremath{\rightarrow}0, throughout the ``Griffiths phase.'' The amplitude A vanishes at the onset of magnetic long-range order, and diverges at the transition between ``Griffiths'' and ``paramagnetic'' phases. For an O(m) spin system, with m\ensuremath{\rightarrow}\ensuremath{\infty}, the spin autocorrelation function C(t) is found to have the ``stretched-exponential'' form, lnC(t)\ensuremath{\sim}-(At)1/2{)}^{1/2})1/2, in the Griffiths phase.
The damping of spin waves in bond-diluted Heisenberg antiferromagnets in d dimensions is considered using a method developed for the ferromagnetic case by the authors. They treat the hydrodynamic limit q xi identical to (k- pi ) xi <<1 and derive an expression for the damping Gamma c (q, xi ) varies as (p-p c ) - mu q d+1 , where mu = nu d- 1 / 2 (2t- beta -(d-2) nu ) which is consistent with the dynamic scaling principle. They also obtain a closed-form expression for the response function and consider finite-cluster response. The critical dynamic properties of the corresponding site-diluted system are then examined. Finally, they contrast d=1 and d>1 and then generalise the results to the ferrimagnet.
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).