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arXiv:cond-mat/9910232 · Journal of Physics A Mathematical and General

Off-equilibrium dynamics at very low temperatures in three-dimensional spin glasses

Enzo Marinari, Giorgio Parisi A, Federico Ricci‐Tersenghi, Juan Jesus Ruiz-Lorenzo

Published Mar 23, 2000• 51 citations
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Abstract

We present a high-statistic systematic study of the overlap correlation function well below the critical temperature in the three-dimensional Gaussian spin glass. The off-equilibrium correlation function has been studied confirming the power law behaviour for the dynamical correlation length. In particular, we have computed the dynamical critical exponent z in a wide range of temperatures, 0.35 T 0.9, obtaining a dependence z ( T ) = 6.2/ T in very good agreement with recent experiments. Moreover, we report a study of the violation of the fluctuation-dissipation theorem for very low temperatures T = 0.5 and 0.35. All our numerical results avoid a droplet model interpretation even when T is as low as T = 0.35.

DOI
10.1088/0305-4470/33/12/305

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We present a high-statistic systematic study of the overlap correlation function well below the critical temperature in the three-dimensional Gaussian spin glass. The off-equilibrium correlation function has been studied confirming the power law behaviour for the dynamical correlation length. In particular, we have computed the dynamical critical exponent z in a wide range of temperatures, 0.35 T 0.9, obtaining a dependence z ( T ) = 6.2/ T in very good agreement with recent experiments. Moreover, we report a study of the violation of the fluctuation-dissipation theorem for very low temperatures T = 0.5 and 0.35. All our numerical results avoid a droplet model interpretation even when T is as low as T = 0.35.
We characterize numerically the properties of the phase transition of the three-dimensional Ising spin glass with Gaussian couplings and of the low-temperature phase. We compute critical exponents on large lattices. We study in detail the overlap probability distribution and the equilibrium overlap-overlap correlation functions. We find a clear agreement with off-equilibrium results from previous work. These results strongly support the existence of a continuous spontaneous replica symmetry breaking in three-dimensional spin glasses.
I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈Sx{S}_{x}Sx​(0)Sx{S}_{x}Sx​(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.
Using the decay of the out equilibrium spin-spin correlation function we compute the equilibrium Edward-Anderson order parameter in the three dimensional binary Ising spin glass in the spin glass phase. We have checked that the Edward-Anderson order parameter computed from out of equilibrium numerical simulations follows with good precision the critical law as determined in experiments and in numerical studies at equilibrium. We have also studied the dependence of the order parameter with the lattice size. Finally we present a large time study of the scaling of the off-equilibrium fluctuation-dissipation relations.
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Some recent experiments on the non‐equilibrium dynamics in two‐ and three‐dimensional spin glasses are reviewed. The experimental results are discussed within new domain (droplet) theories of the spin glass phase. It is found that the experiments convincingly verify some fundamental concepts of these theories. Especially, it is shown that if a spin glass has been ‘‘aged’’ at constant temperature and afterwards subjected to a persistent temperature shift or a temperature cycling, the dynamics are governed by an interplay between the growth of spin glass domains and the overlap length, which is the maximum length scale where equilibrium spin correlations at two nearby temperatures are indistinguishable.
Dynamical scaling study for the estimation of dynamical exponent zzz of three-dimensional XY spin glass model
To analyze the ±J\pm J±J XY spin-glass in three dimensions, we verified a method aimed at obtaining a high-precision dynamical exponent zzz from the correlation length in the nonequilibrium relaxation process. The obtained zzz yielded consistent and highly accurate results in previous studies for relatively well-studied models---specifically, the three-dimensional (3D) ferromagnetic Ising model and the 3D ±J\pm J±J Ising model. Building on these previous studies, we used this method and the dynamical scaling method to analyze the 3D ±J\pm J±J XY model and obtained highly precise critical temperatures and exponents. The estimated transition temperatures satisfy $T_{\mathrm{SG}}