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Physical Review Letters

Dynamics of dilute magnets above Tc

Alan J. Bray

Published Feb 22, 1988• 101 citations
Publisher

Abstract

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{-${\mathit{t}}^{\mathit{d}/(\mathit{d}+\mathit{z})}$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${)}^{1/2}$} are predicted for Ising Heisenberg systems, respectively, for t\ensuremath{\gg}${\ensuremath{\xi}}_{\mathit{p}}^{\mathit{d}+\mathit{z}}$.

DOI
10.1103/physrevlett.60.720

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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/\ensuremathξpd+z{\ensuremath{\xi}}_{\mathit{p}}^{\mathit{d}+\mathit{z}}\ensuremathξpd+z​)} is predicted for the spin autocorrelation function, with \ensuremathξp{\ensuremath{\xi}}_{\mathit{p}}\ensuremathξp​ the correlation length of the nondilute system. In the Griffiths phase, C(t)\ensuremath{\sim}exp{-A1nt)d/(d\ensuremath−1){)}^{\mathit{d}/(\mathit{d}\mathrm{\ensuremath{-}}1)})d/(d\ensuremath−1)} and exp{-(Bt)1/2{)}^{1/2})1/2} are predicted for Ising Heisenberg systems, respectively, for t\ensuremath{\gg}\ensuremathξpd+z{\ensuremath{\xi}}_{\mathit{p}}^{\mathit{d}+\mathit{z}}\ensuremathξpd+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.
A soluble, but nontrivial, model of a dilute Ising ferromagnet is studied, with infinite-range interactions but finite average connectivity c. The density of (Yang-Lee) zeros of the partition function in the complex z=exp(-2H) plane (where H is the external magnetic field in units of the temperature) is calculated explicitly in the high-temperature phase for large but finite c and small \ensuremath{\Vert}H\ensuremath{\Vert}. The density of zeros on the unit circle H=i\ensuremath{\theta} has the form \ensuremath{\rho}(\ensuremath{\theta})\ensuremath{\sim}exp{-[cf(K)/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}]ln/B (1/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert})} for \ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rightarrow}0. The function f(K) (K=J/T) vanishes at the critical coupling KC{K}_{C}KC​(c). Heuristic arguments are given for the form of \ensuremath{\rho}(\ensuremath{\theta}) expected for systems with short-range interactions.
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 .