Physical review. B, Condensed matter

Nonequilibrium dynamics of spin glasses

Daniel S. Fisher, David A. Huse

Published Jul 1, 1988• 824 citations

Abstract

We consider the nonequilibrium behavior of the spin-glass ordered phase within the droplet scaling theory introduced previously. The fundamental long-time nonequilibrium process is assumed to be the thermally activated growth of spin-glass ordered domains. The remanent magnetization, m(t), in zero field is found to decay at long times as m(t)\ensuremath{\sim}${R}_{t}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$, where ${R}_{t}$\ensuremath{\sim}(lnt${)}^{1/\ensuremath{\psi}}$ is the linear domain size, \ensuremath{\psi} is the previously introduced barrier exponent describing the growth of activation-barrier heights with length scale, and \ensuremath{\lambda} is a new nonequilibrium dynamic exponent, satisfying the relation \ensuremath{\lambda}\ensuremath{\ge}d/2 for d-dimensional systems. The effects of waiting for partial equilibration before making a measurement are studied in various regimes. The effects of quenching first to one temperature and then to another are also examined. Such experiments can, in principle, be used to obtain information about the relative rate of dynamic evolution as well as the overlap between the equilibrium states at different temperatures. In particular, the length scale ${L}_{\ensuremath{\Delta}T}$, below which equilibrium correlations at temperatures T and T+\ensuremath{\Delta}T are similar, plays an important role. The decay of m(t) and the growth of spin-glass order after a quench are examined in Monte Carlo simulations of the Sherrington-Kirkpatrick model.

DOI: 10.1103/physrevb.38.373

Related papers

Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin glass

Cécile Monthus, Thomas Garel

arXiv:1309.2154Physical Review BJan 9, 201411 citations

Numerical study of the dynamics of some long range spin glass models

Alain H. Billoire

arXiv:1505.00676Journal of Statistical Mechanics Theory and ExperimentJul 30, 20152 citations

Dynamics of three-dimensional Ising spin glasses in thermal equilibrium

Andrew T. Ogielski

Physical review. B, Condensed matterDec 1, 1985773 citations

Understanding Nonequilibrium Behaviors of Spin Glasses Through Heuristics

Jie Yang

Jan 1, 20180 citations

An experiment-oriented analysis of 2D spin-glass dynamics: a twelve time-decades scaling study

L A Fernandez, Enzo Marinari, V Martin-Mayor +2 more

arXiv:1805.06738Journal of Physics A Mathematical and TheoreticalMar 26, 201914 citations

Monte Carlo studies of Ising spin-glass systems: Aging behavior and crossover between equilibrium and nonequilibrium dynamics

J.-O. Andersson, Johan Mattsson, Peter Svedlindh

Physical review. B, Condensed matterOct 1, 199265 citations

Physical review. B, Condensed matter

Nonequilibrium dynamics of spin glasses

Daniel S. Fisher, David A. Huse

Published Jul 1, 1988• 824 citations

Publisher

Abstract

{R}_{t}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}

, where

{R}_{t}

\ensuremath{\sim}(lnt

{)}^{1/\ensuremath{\psi}}

is the linear domain size, \ensuremath{\psi} is the previously introduced barrier exponent describing the growth of activation-barrier heights with length scale, and \ensuremath{\lambda} is a new nonequilibrium dynamic exponent, satisfying the relation \ensuremath{\lambda}\ensuremath{\ge}d/2 for d-dimensional systems. The effects of waiting for partial equilibration before making a measurement are studied in various regimes. The effects of quenching first to one temperature and then to another are also examined. Such experiments can, in principle, be used to obtain information about the relative rate of dynamic evolution as well as the overlap between the equilibrium states at different temperatures. In particular, the length scale

{L}_{\ensuremath{\Delta}T}

, below which equilibrium correlations at temperatures T and T+\ensuremath{\Delta}T are similar, plays an important role. The decay of m(t) and the growth of spin-glass order after a quench are examined in Monte Carlo simulations of the Sherrington-Kirkpatrick model.

DOI: 10.1103/physrevb.38.373

Related papers

Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin glass

Cécile Monthus, Thomas Garel

The long-range one-dimensional Ising spin glass with random couplings decaying as $J(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\sigma}}$ presents a spin-glass phase ${T}_{c}(\ensuremath{\sigma})>0$ for $0\ensuremath{\le}\ensuremath{\sigma}<1$ (the limit $\ensuremath{\sigma}=0$ corresponds to the mean-field Sherrington-Kirkpatrick model). We use the eigenvalue method introduced in our previous work (C. Monthus and T. Garel, J. Stat. Mech. 2009, P12017) to measure the equilibrium time ${t}_{eq}(N)$ at temperature $T={T}_{c}(\ensuremath{\sigma})/2$ as a function of the number $N$ of spins. We find the activated scaling $\overline{ln{t}_{eq}(N)}\ensuremath{\sim}{N}^{\ensuremath{\psi}}$ with the same barrier exponent $\ensuremath{\psi}\ensuremath{\simeq}0.33$ in the whole region $0\ensuremath{\le}\ensuremath{\sigma}<1$.The long-range one-dimensional Ising spin glass with random couplings decaying as $J(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\sigma}}$ presents a spin-glass phase ${T}_{c}(\ensuremath{\sigma})>0$ for $0\ensuremath{\le}\ensuremath{\sigma}<1$ (the limit $\ensuremath{\sigma}=0$ corresponds to the mean-field Sherrington-Kirkpatrick model). We use the eigenvalue method introduced in our previous work (C. Monthus and T. Garel, J. Stat. Mech. 2009, P12017) to measure the equilibrium time ${t}_{eq}(N)$ at temperature $T={T}_{c}(\ensuremath{\sigma})/2$ as a function of the number $N$ of spins. We find the activated scaling $\overline{ln{t}_{eq}(N)}\ensuremath{\sim}{N}^{\ensuremath{\psi}}$ with the same barrier exponent $\ensuremath{\psi}\ensuremath{\simeq}0.33$ in the whole region $0\ensuremath{\le}\ensuremath{\sigma}<1$.

arXiv:1309.2154Physical Review BJan 9, 201411 citations

Numerical study of the dynamics of some long range spin glass models

Alain H. Billoire

We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter , this model interpolates between the mean field Sherrington–Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling ( ) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington–Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter , this model interpolates between the mean field Sherrington–Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling ( ) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington–Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.

arXiv:1505.00676Journal of Statistical Mechanics Theory and ExperimentJul 30, 20152 citations

Dynamics of three-dimensional Ising spin glasses in thermal equilibrium

Andrew T. Ogielski

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)=〈 ${S}_{x}$ (0) ${S}_{x}$ (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 ( ${8}^{3}$ , ${16}^{3}$ , ${32}^{3}$ , and ${64}^{3}$ ). 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.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)=〈${S}_{x}$(0)${S}_{x}$(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 (${8}^{3}$, ${16}^{3}$, ${32}^{3}$, and ${64}^{3}$). 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.

Physical review. B, Condensed matterDec 1, 1985773 citations

Understanding Nonequilibrium Behaviors of Spin Glasses Through Heuristics

Jie Yang

Spin glass behavior was first seen in metallic alloys with magnetic impurities dispersed randomly in the main non-magnetic component. Due to the simultaneous presence of quenched disorder and competing ferromagnetic and antiferromagnetic interactions, the nonequilibrium behaviors of spin glasses are intricate and difficult to understand. Experimental research on spin glass materials indicate extremely slow dynamics, suggesting that the direct numerical simulation using local spin flips will take impractical time scales to replicate the nonequilibrium behaviors. This thesis introduces efficient algorithms and heuristics for the numerical simulation of simple models for spin glasses and discusses significant simulation results. The results provide insights into understanding the nonequilibrium behaviors, especially aging and memory, through studying microscopic constituent configurations. The methodology and conclusion presented in this thesis can be potentially extended to address general questions about disordered materials at nonequilibrium. Chapter 1 provides a thorough overview of spin glasses. A detailed introduction on complex nonequilibrium behaviors, including slow dynamics and memory effects, seen in typical experiments of spin glass materials, is presented. A general discussion about the characterization and categorization of memory effects found in many out-of-equilibrium systems is highlighted. Two prevalent theoretical models on spin glasses, the droplet and replica symmetry breaking theory, are described. Lastly, algorithm development and simulation techniques used to study spin glasses are covered. In Chapter 2, a patchwork heuristic is applied to accelerate the relaxation of spin glass models in dimensions $d=1$ and $d=2$ with the goal of investigating the dynamics of loss and recovery of spin configurations. A well-designed simulation protocol, consists of a aging and recovery stage, is used to mimic the temperature cycling experiment, where the temperature variations are replaced by the variations of the spin coupling realization. Eventual recovery of memory upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems do not exhibit configuration memory. A central result is that the coarsening recovery length scale $s_{\mathrm c}$ grows faster than the aging length scale $\ell$ as a power law relation $s_{\mathrm c}\sim\ell^{2.58}$ . In Chapter 3, the microscopic dependence of spin glass equilibration on sample history, i.e., the initial configuration and the specific noise history of stochastic dynamics, is examined. The patchwork dynamics is used to simulate the glassy dynamics over a wide range of length scales. The dependence of the nearest-neighbor spin correlations at long time on initial spin configurations and on noise histories are compared. Most of the local spin correlations are independent of the details of evolution, due to the formation of rigid domains. The correlations on a fractal set of domain walls are found to have distinct dependence on the noise history and the initial state. The initial ``nature'' of the state is retained in the domain walls under coarsening more than the ``nurture'' of the noise, while the location of the domain walls at each scale is determined by the frozen disorder (``environment''). We also provide evidence that coarsening with local dynamics at finite temperature gives persistence of overlaps roughly consistent with coarsening at zero temperature, at least over short distances, supporting the use of zero-temperature heuristics to address finite-temperature dynamics and memory.Spin glass behavior was first seen in metallic alloys with magnetic impurities dispersed randomly in the main non-magnetic component. Due to the simultaneous presence of quenched disorder and competing ferromagnetic and antiferromagnetic interactions, the nonequilibrium behaviors of spin glasses are intricate and difficult to understand. Experimental research on spin glass materials indicate extremely slow dynamics, suggesting that the direct numerical simulation using local spin flips will take impractical time scales to replicate the nonequilibrium behaviors. This thesis introduces efficient algorithms and heuristics for the numerical simulation of simple models for spin glasses and discusses significant simulation results. The results provide insights into understanding the nonequilibrium behaviors, especially aging and memory, through studying microscopic constituent configurations. The methodology and conclusion presented in this thesis can be potentially extended to address general questions about disordered materials at nonequilibrium. Chapter 1 provides a thorough overview of spin glasses. A detailed introduction on complex nonequilibrium behaviors, including slow dynamics and memory effects, seen in typical experiments of spin glass materials, is presented. A general discussion about the characterization and categorization of memory effects found in many out-of-equilibrium systems is highlighted. Two prevalent theoretical models on spin glasses, the droplet and replica symmetry breaking theory, are described. Lastly, algorithm development and simulation techniques used to study spin glasses are covered. In Chapter 2, a patchwork heuristic is applied to accelerate the relaxation of spin glass models in dimensions $d=1$ and $d=2$ with the goal of investigating the dynamics of loss and recovery of spin configurations. A well-designed simulation protocol, consists of a aging and recovery stage, is used to mimic the temperature cycling experiment, where the temperature variations are replaced by the variations of the spin coupling realization. Eventual recovery of memory upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems do not exhibit configuration memory. A central result is that the coarsening recovery length scale $s_{\mathrm c}$ grows faster than the aging length scale $\ell$ as a power law relation $s_{\mathrm c}\sim\ell^{2.58}$. In Chapter 3, the microscopic dependence of spin glass equilibration on sample history, i.e., the initial configuration and the specific noise history of stochastic dynamics, is examined. The patchwork dynamics is used to simulate the glassy dynamics over a wide range of length scales. The dependence of the nearest-neighbor spin correlations at long time on initial spin configurations and on noise histories are compared. Most of the local spin correlations are independent of the details of evolution, due to the formation of rigid domains. The correlations on a fractal set of domain walls are found to have distinct dependence on the noise history and the initial state. The initial ``nature'' of the state is retained in the domain walls under coarsening more than the ``nurture'' of the noise, while the location of the domain walls at each scale is determined by the frozen disorder (``environment''). We also provide evidence that coarsening with local dynamics at finite temperature gives persistence of overlaps roughly consistent with coarsening at zero temperature, at least over short distances, supporting the use of zero-temperature heuristics to address finite-temperature dynamics and memory.

Jan 1, 20180 citations

An experiment-oriented analysis of 2D spin-glass dynamics: a twelve time-decades scaling study

L A Fernandez, Enzo Marinari, V Martin-Mayor +2 more

Abstract Recent high precision experimental results on spin-glass films ask for a detailed understanding of the domain-growth dynamics of two-dimensional spin glasses. To achieve this goal, we numerically simulate the out-equilibrium dynamics of the Ising spin glass for a time that spans close to twelve orders of magnitude (from picoseconds to order of a second), in systems large enough to avoid finite-size effects. We find that the time-growth of the size of the glassy domains is excellently described by a single scaling function. A single time-scale controls the dynamics. diverges upon approaching the T = 0 critical point. The divergence of is Arrhenius-like, with a barrier height that depends very mildly on temperature. The growth of this barrier-height is best described by critical dynamics. As a side product we obtain an impressive confirmation of universality of the equilibrium behavior of two-dimensional spin-glasses.Abstract Recent high precision experimental results on spin-glass films ask for a detailed understanding of the domain-growth dynamics of two-dimensional spin glasses. To achieve this goal, we numerically simulate the out-equilibrium dynamics of the Ising spin glass for a time that spans close to twelve orders of magnitude (from picoseconds to order of a second), in systems large enough to avoid finite-size effects. We find that the time-growth of the size of the glassy domains is excellently described by a single scaling function. A single time-scale controls the dynamics. diverges upon approaching the T = 0 critical point. The divergence of is Arrhenius-like, with a barrier height that depends very mildly on temperature. The growth of this barrier-height is best described by critical dynamics. As a side product we obtain an impressive confirmation of universality of the equilibrium behavior of two-dimensional spin-glasses.

arXiv:1805.06738Journal of Physics A Mathematical and TheoreticalMar 26, 201914 citations

Monte Carlo studies of Ising spin-glass systems: Aging behavior and crossover between equilibrium and nonequilibrium dynamics

J.-O. Andersson, Johan Mattsson, Peter Svedlindh

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 ${\mathit{t}}_{\mathit{w}}$ , 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} ${\mathrm{log}}_{10}$ (t) exhibits a maximum at t\ensuremath{\approxeq} ${\mathit{t}}_{\mathit{w}}$ . If the temperature is lowered or raised, immediately prior to the field application, the apparent age ( ${\mathit{t}}_{\mathit{a}}$ ) 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} ${\mathit{t}}_{\mathit{w}}$ ), clear deviations are found at longer time scales (t> ${\mathit{t}}_{\mathit{w}}$ ). 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.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 ${\mathit{t}}_{\mathit{w}}$, 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} ${\mathrm{log}}_{10}$(t) exhibits a maximum at t\ensuremath{\approxeq}${\mathit{t}}_{\mathit{w}}$. If the temperature is lowered or raised, immediately prior to the field application, the apparent age (${\mathit{t}}_{\mathit{a}}$) 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}${\mathit{t}}_{\mathit{w}}$), clear deviations are found at longer time scales (t>${\mathit{t}}_{\mathit{w}}$). 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.

Physical review. B, Condensed matterOct 1, 199265 citations

{R}_{t}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}

, where

{R}_{t}

\ensuremath{\sim}(lnt

{)}^{1/\ensuremath{\psi}}

{L}_{\ensuremath{\Delta}T}

{t}_{eq}(N)

at temperature $T={T}_{c}(\ensuremath{\sigma})/2$ as a function of the number

N

of spins. We find the activated scaling $\overline{ln{t}_{eq}(N)}\ensuremath{\sim}{N}^{\ensuremath{\psi}}$ with the same barrier exponent $\ensuremath{\psi}\ensuremath{\simeq}0.33$ in the whole region $0\ensuremath{\le}\ensuremath{\sigma}<1$.

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)=〈

{S}_{x}

(0)

{S}_{x}

(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 (

{8}^{3}

{16}^{3}

{32}^{3}

, and

{64}^{3}

). 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.

d=1

and

d=2

with the goal of investigating the dynamics of loss and recovery of spin configurations. A well-designed simulation protocol, consists of a aging and recovery stage, is used to mimic the temperature cycling experiment, where the temperature variations are replaced by the variations of the spin coupling realization. Eventual recovery of memory upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems do not exhibit configuration memory. A central result is that the coarsening recovery length scale

s_{\mathrm c}

grows faster than the aging length scale

\ell

as a power law relation

s_{\mathrm c}\sim\ell^{2.58}

. In Chapter 3, the microscopic dependence of spin glass equilibration on sample history, i.e., the initial configuration and the specific noise history of stochastic dynamics, is examined. The patchwork dynamics is used to simulate the glassy dynamics over a wide range of length scales. The dependence of the nearest-neighbor spin correlations at long time on initial spin configurations and on noise histories are compared. Most of the local spin correlations are independent of the details of evolution, due to the formation of rigid domains. The correlations on a fractal set of domain walls are found to have distinct dependence on the noise history and the initial state. The initial ``nature'' of the state is retained in the domain walls under coarsening more than the ``nurture'' of the noise, while the location of the domain walls at each scale is determined by the frozen disorder (``environment''). We also provide evidence that coarsening with local dynamics at finite temperature gives persistence of overlaps roughly consistent with coarsening at zero temperature, at least over short distances, supporting the use of zero-temperature heuristics to address finite-temperature dynamics and memory.

{\mathit{t}}_{\mathit{w}}

, 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}

{\mathrm{log}}_{10}

(t) exhibits a maximum at t\ensuremath{\approxeq}

{\mathit{t}}_{\mathit{w}}

. If the temperature is lowered or raised, immediately prior to the field application, the apparent age (

{\mathit{t}}_{\mathit{a}}

) 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}

{\mathit{t}}_{\mathit{w}}

), clear deviations are found at longer time scales (t>

{\mathit{t}}_{\mathit{w}}

). 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.