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Physical review. B, Condensed matter

Dynamics of random ising ferromagnets in the griffiths phase

Alan J. Bray, Geoff J. Rodgers

Published Nov 1, 1988• 25 citations
Publisher

Abstract

The dynamics of a random Ising ferromagnet is considered, for a general distribution $P(J)$ of exchange interactions, in the Griffiths phase. The autocorrelation function $C(t)$ 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 $d$ 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 $C(t)$ depends explicitly on the form of $P(J)$ at large $J$.

DOI
10.1103/physrevb.38.9252

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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∼exp[\ensuremath−A(lnt)d(d\ensuremath−1)]C(t)\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}A{(\mathrm{ln}t)}^{\frac{d}{(d\ensuremath{-}1)}}]C(t)\ensuremath∼exp[\ensuremath−A(lnt)(d\ensuremath−1)d​], where ddd is the spatial dimension, only if \ensuremath−ln[P(J)]\ensuremath{-}\mathrm{ln}[P(J)]\ensuremath−ln[P(J)] grows faster than Jd(d\ensuremath−1){J}^{\frac{d}{(d\ensuremath{-}1)}}J(d\ensuremath−1)d​ as J\ensuremath→\ensuremath∞J\ensuremath{\rightarrow}\ensuremath{\infty}J\ensuremath→\ensuremath∞. 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.
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 stochastic dynamics of randomly asymmetric fully connected Ising systems is studied. We solve analytically the particularly simple case of fully asymmetric systems. We calculate the relaxation time of the autocorrelation function and show that the system remains paramagnetic even at zero temperature (T=0). The ferromagnetic phase is only slightly affected by the asymmetry, and the paramagnetic-to-ferromagnetic phase transition is characterized by a critical slowing down similar to second-order transition in symmetric (fully connected) systems. Monte Carlo simulations of a fully connected Ising system with random asymmetric interactions, both at finite and zero temperature, are presented. For finite T the autocorrelation function decays completely to zero for all strengths of the asymmetry. The T=0 behavior is more complex. In the fully asymmetric case the system is ergodic, with decaying autocorrelations, in agreement with the theoretical predictions. In the partially asymmetric case all flows terminate at fixed points (i.e., states which are stable to single spin flips). However, the typical time that it takes to converge to a fixed point grows exponentially with the size of the system. This convergence time varies from sample to sample and has a log-normal distribution in large systems. On time scales which are smaller than the convergence time, the system behaves ``ergodically,'' and the autocorrelation function decays to zero, much like the finite-temperature case.
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.
We consider the paramagnetic phase of the random transverse-field Ising spin chain and study the dynamical properties by numerical methods and scaling considerations. We extend our previous work [Phys. Rev. B 57, 11 404 (1998)] to new quantities, such as the nonlinear susceptibility, higher excitations, and the energy-density autocorrelation function. We show that in the Griffiths phase all the above quantities exhibit power-law singularities and the corresponding critical exponents, which vary with the distance from the critical point, can be related to the dynamical exponent z, the latter being the positive root of [(J/h)1/z]av=1[{(J/h)}^{1/z}{]}_{\mathrm{av}}=1[(J/h)1/z]av​=1. Particularly, whereas the average spin autocorrelation function in imaginary time decays as $[G{]}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/z}$, the average energy-density autocorrelations decay with another exponent as $[{G}^{e}{]}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}2\ensuremath{-}1/z}$.