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Physical review. B, Condensed matter

Nonequilibrium relaxation study of Ising spin glass models

Yukiyasu Ozeki, Nobuyasu Ito

Published Jun 20, 2001• 36 citations
Publisher

Abstract

As an analysis of equilibrium phase transitions, the nonequilibrium relaxation method is extended to the spin glass (SG) transition. The $\ifmmode\pm\else\textpm\fi{}J$ Ising SG model is analyzed for three-dimensional (cubic) lattices up to the linear size of $L=127$ and for four-dimensional (hypercubic) lattice up to $L=41.$ These sizes of systems are quite large as compared with those calculated, so far, by equilibrium simulations. As a dynamical order parameter, we calculate the clone correlation function (CCF) ${Q(t,t}_{\mathrm{w}})\ensuremath{\equiv}[〈{S}_{i}^{(1)}{(t+t}_{\mathrm{w}}){S}_{i}^{(2)}{(t+t}_{\mathrm{w}}){〉}^{\mathrm{F}}],$ which is a spin correlation of two replicas produced after the waiting time ${t}_{\mathrm{w}}$ from a simple starting state. It is found that the CCF shows an exponential decay in the paramagnetic phase, and a power-law decay after aginglike development $(t\ensuremath{\gg}{t}_{\mathrm{w}})$ in the SG phase. This provides a reliable upper bound of the transition temperature ${T}_{\mathrm{g}}.$ It is also found that a scaling relation, ${Q(t,t}_{\mathrm{w}}{)=t}_{\mathrm{w}}^{\ensuremath{-}{\ensuremath{\lambda}}_{q}}\overline{q}{(t/t}_{\mathrm{w}}),$ holds just around the transition point providing the lower bound of ${T}_{\mathrm{g}}.$ Together with these two bounds, we propose a new dynamical way for the estimation of ${T}_{\mathrm{g}}$ from much larger systems. In the SG phase, the power-law behavior of the CCF for $t\ensuremath{\gg}{t}_{\mathrm{w}}$ suggests that the SG phase in short-range Ising models has a rugged phase space.

DOI
10.1103/physrevb.64.024416

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As an analysis of equilibrium phase transitions, the nonequilibrium relaxation method is extended to the spin glass (SG) transition. The \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ Ising SG model is analyzed for three-dimensional (cubic) lattices up to the linear size of L=127L=127L=127 and for four-dimensional (hypercubic) lattice up to L=41.L=41.L=41. These sizes of systems are quite large as compared with those calculated, so far, by equilibrium simulations. As a dynamical order parameter, we calculate the clone correlation function (CCF) Q(t,tw)\ensuremath≡[〈Si(1)(t+tw)Si(2)(t+tw)〉F],{Q(t,t}_{\mathrm{w}})\ensuremath{\equiv}[〈{S}_{i}^{(1)}{(t+t}_{\mathrm{w}}){S}_{i}^{(2)}{(t+t}_{\mathrm{w}}){〉}^{\mathrm{F}}],Q(t,tw​)\ensuremath≡[〈Si(1)​(t+tw​)Si(2)​(t+tw​)〉F], which is a spin correlation of two replicas produced after the waiting time tw{t}_{\mathrm{w}}tw​ from a simple starting state. It is found that the CCF shows an exponential decay in the paramagnetic phase, and a power-law decay after aginglike development (t\ensuremath≫tw)(t\ensuremath{\gg}{t}_{\mathrm{w}})(t\ensuremath≫tw​) in the SG phase. This provides a reliable upper bound of the transition temperature Tg.{T}_{\mathrm{g}}.Tg​. It is also found that a scaling relation, Q(t,tw)=tw\ensuremath−\ensuremathλqq‾(t/tw),{Q(t,t}_{\mathrm{w}}{)=t}_{\mathrm{w}}^{\ensuremath{-}{\ensuremath{\lambda}}_{q}}\overline{q}{(t/t}_{\mathrm{w}}),Q(t,tw​)=tw\ensuremath−\ensuremathλq​​q​(t/tw​), holds just around the transition point providing the lower bound of Tg.{T}_{\mathrm{g}}.Tg​. Together with these two bounds, we propose a new dynamical way for the estimation of Tg{T}_{\mathrm{g}}Tg​ from much larger systems. In the SG phase, the power-law behavior of the CCF for t\ensuremath≫twt\ensuremath{\gg}{t}_{\mathrm{w}}t\ensuremath≫tw​ suggests that the SG phase in short-range Ising models has a rugged phase space.
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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{\mathit{t}}_{\mathit{w}}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{\mathrm{log}}_{10}log10​(t) exhibits a maximum at t\ensuremath{\approxeq}tw{\mathit{t}}_{\mathit{w}}tw​. If the temperature is lowered or raised, immediately prior to the field application, the apparent age (ta{\mathit{t}}_{\mathit{a}}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{\mathit{t}}_{\mathit{w}}tw​), clear deviations are found at longer time scales (t>tw{\mathit{t}}_{\mathit{w}}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.
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