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Physical Review Letters

Domain growth by aging in nonequilibrium two-dimensional random Ising systems

A. G. Schins, A. F. M. Arts, H. W. de Wijn

Published Apr 12, 1993• 51 citations
Publisher

Abstract

The ac susceptibility in the two-dimensional Ising spin glass ${\mathrm{Rb}}_{2}$${\mathrm{Cu}}_{0.67}$${\mathrm{Co}}_{0.33}$${\mathrm{F}}_{4}$ and the two-dimensional random-exchange Ising ferromagnet ${\mathrm{Rb}}_{2}$${\mathrm{Cu}}_{0.89}$${\mathrm{Co}}_{0.11}$${\mathrm{F}}_{4}$ is observed to relax towards equilibrium on a time scale of hours following fast quenching to 3.9 K. Scaling of the in-phase and out-of-phase susceptibilities yields that the ordered domains grow according to (lnt${)}^{1/\mathrm{\ensuremath{\psi}}}$, with t expressed in units of a microscopic time, and \ensuremath{\psi}=1.07\ifmmode\pm\else\textpm\fi{}0.10 and 0.20\ifmmode\pm\else\textpm\fi{}0.05, respectively. Both values are in agreement with recent theoretical results and numerical simulations.

DOI
10.1103/physrevlett.70.2340

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Aging of two-dimensional (2D) Ising spin glasses with short-range interactions ${\mathrm{Rb}}_{2}$${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Co}}_{\mathit{x}}$${\mathrm{F}}_{4}$ (${\mathit{T}}_{\mathit{c}}$=0) is studied in the critical regime via the low-frequency ac susceptibility and via the dc magnetization for x=0.22 and 0.33. After quenching to low temperatures, slow logarithmic relaxation is observed for both the in-phase and out-of-phase ac susceptibilities. Using the droplet scaling model, the spin-glass correlation length R is concluded to increase after a temperature quench as R\ensuremath{\propto}(lnt${)}^{1/\mathrm{\ensuremath{\psi}}}$, with t in units of some microscopic time. The exponent \ensuremath{\psi}=1.0\ifmmode\pm\else\textpm\fi{}0.1 compares with recent theoretical results and numerical simulations for 2D short-range Ising spin glasses. The waiting-time dependence found for the field-cooled and zero-field-cooled dc magnetizations points to a markedly slower crossover from the quasiequilibrium to the nonequilibrium regime than in 3D. From experiments with different cooling histories, finally, the equilibrium spin-glass state is inferred to be strongly dependent on the temperature.

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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 ${\mathit{t}}_{\mathit{w}}$, 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.

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Spin Systems far from Equilibrium: Aging and Dynamic Phase Transition

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Among the many non-equilibrium processes encountered in nature we deal with two different but related aspects. One is the non-equilibrium relaxation process that is at the origin of \'aging phenomena••, and the other one is a non-equilibrium phase transition, called ••dynamic phase transition••. One of the main purposes of our research is to explore more realistic situations than studied previously. Indeed, in the study of aging phenomena certain kinds of disorder effects are considered, and we introduce the ••surface•• as a spatial boundary to the system undergoing the dynamic phase transition. In order to observe these processes as clearly as possible, we study in both cases simple spin systems. Using Monte Carlo simulations we first investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length L(t) is determined numerically and the scaling behavior of various two-time quantities as a function of L(t)/L(s) is discussed where t and s are two different times. For disordered Ising models deviations of L(t) from algebraic growth law show up. The generalized scaling forms as a function of L(t)/L(s) reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets. We also study the local critical phenomena at a dynamic phase transition by means of numerical simulations of kinetic Ising models with surfaces subjected to a periodic oscillating field. We examine layer-dependent quantities, such as the period-averaged magnetization per layer Q(z) and the layer susceptibility ¥ö(z), and determine local critical exponents through finite size scaling. Both for two and three dimensions, we find that the values of the surface exponents differ from those of the equilibrium critical surface. It is revealed that the surface phase diagram of the non-equilibrium system is not identical to that of the equilibrium system in three dimensions.

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By means of Monte Carlo simulations on the three-dimensional Ising spin-glass model, we have studied aging phenomena with various temperature($T$)-change protocols. Particularly, a $T$-shift protocol, in which a system is first quenched to and aged for a period $t_{\rm w1}$ at a temperature $T_1$, and subsequently aged at a new temperature $T_2$ is closely investigated. Most importantly, the mean size of domains, which is extracted from the replica-overlap function, was found to grow monotonically without any appreciable decrease by the $T$-change. We also found the relaxation of energy density and spin auto-correlation function during the $T$-shift process can be explained within the following picture: the dynamics finally crossovers to an isothermal aging of $T_2$ at around $t_2 \simeq t_{\rm w1}^{\rm (eff)}$ after the $T$-change, where $t_{\rm w1}^{\rm (eff)}$ is an effective waiting time related with $t_{\rm w1}$, but in the transient regime $t_2 \lt t_{\rm w1}^{\rm (eff)}$, adjustment of the population of thermally active droplets from that at $T_1$ to $T_2$ takes place very slowly. Implications of the results on aging phenomena in other $T$-change protocols such as $T$-cycling and continuous $T$-change with an intermittent stop observed by simulations as well as experiments are discussed.

arXiv:cond-mat/0001395•Jan 27, 2000•7 citations
The ac susceptibility in the two-dimensional Ising spin glass {\mathrm{Rb}}_{2}$${\mathrm{Cu}}_{0.67}$${\mathrm{Co}}_{0.33}$${\mathrm{F}}_{4} and the two-dimensional random-exchange Ising ferromagnet {\mathrm{Rb}}_{2}$${\mathrm{Cu}}_{0.89}$${\mathrm{Co}}_{0.11}$${\mathrm{F}}_{4} is observed to relax towards equilibrium on a time scale of hours following fast quenching to 3.9 K. Scaling of the in-phase and out-of-phase susceptibilities yields that the ordered domains grow according to (lnt)1/\ensuremathψ{)}^{1/\mathrm{\ensuremath{\psi}}})1/\ensuremathψ, with t expressed in units of a microscopic time, and \ensuremath{\psi}=1.07\ifmmode\pm\else\textpm\fi{}0.10 and 0.20\ifmmode\pm\else\textpm\fi{}0.05, respectively. Both values are in agreement with recent theoretical results and numerical simulations.
Aging of two-dimensional (2D) Ising spin glasses with short-range interactions Rb2{\mathrm{Rb}}_{2}Rb2​${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$Cox{\mathrm{Co}}_{\mathit{x}}Cox​F4{\mathrm{F}}_{4}F4​ (Tc{\mathit{T}}_{\mathit{c}}Tc​=0) is studied in the critical regime via the low-frequency ac susceptibility and via the dc magnetization for x=0.22 and 0.33. After quenching to low temperatures, slow logarithmic relaxation is observed for both the in-phase and out-of-phase ac susceptibilities. Using the droplet scaling model, the spin-glass correlation length R is concluded to increase after a temperature quench as R\ensuremath{\propto}(lnt${)}^{1/\mathrm{\ensuremath{\psi}}}$, with t in units of some microscopic time. The exponent \ensuremath{\psi}=1.0\ifmmode\pm\else\textpm\fi{}0.1 compares with recent theoretical results and numerical simulations for 2D short-range Ising spin glasses. The waiting-time dependence found for the field-cooled and zero-field-cooled dc magnetizations points to a markedly slower crossover from the quasiequilibrium to the nonequilibrium regime than in 3D. From experiments with different cooling histories, finally, the equilibrium spin-glass state is inferred to be strongly dependent on the temperature.
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.
Among the many non-equilibrium processes encountered in nature we deal with two different but related aspects. One is the non-equilibrium relaxation process that is at the origin of \'aging phenomena••, and the other one is a non-equilibrium phase transition, called ••dynamic phase transition••. One of the main purposes of our research is to explore more realistic situations than studied previously. Indeed, in the study of aging phenomena certain kinds of disorder effects are considered, and we introduce the ••surface•• as a spatial boundary to the system undergoing the dynamic phase transition. In order to observe these processes as clearly as possible, we study in both cases simple spin systems. Using Monte Carlo simulations we first investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length L(t) is determined numerically and the scaling behavior of various two-time quantities as a function of L(t)/L(s) is discussed where t and s are two different times. For disordered Ising models deviations of L(t) from algebraic growth law show up. The generalized scaling forms as a function of L(t)/L(s) reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets. We also study the local critical phenomena at a dynamic phase transition by means of numerical simulations of kinetic Ising models with surfaces subjected to a periodic oscillating field. We examine layer-dependent quantities, such as the period-averaged magnetization per layer Q(z) and the layer susceptibility ¥ö(z), and determine local critical exponents through finite size scaling. Both for two and three dimensions, we find that the values of the surface exponents differ from those of the equilibrium critical surface. It is revealed that the surface phase diagram of the non-equilibrium system is not identical to that of the equilibrium system in three dimensions.
Using Monte Carlo simulations, we have studied isothermal aging of three-dimensional Ising spin-glass model focusing on quasi-equilibrium behavior of the spin auto-correlation function. Weak violation of the time translational invariance in the quasi-equilibrium regime is analyzed in terms of {\it effective stiffness} for droplet excitations in the presence of domain walls. Within the range of computational time window, we have confirmed that the effective stiffness follows the expected scaling behavior with respect to the characteristic length scales associated with droplet excitations and domain walls, whose growth law has been extracted from our simulated data. Implication of the results are discussed in relation to experimental works on ac susceptibilities.
We have studied numerically the dynamics of spin glasses with Ising and XYXYXY symmetry (gauge glass) in space dimensions 2, 3, and 4. The nonequilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)$ and the nonequilibrium energy per spin, ene(tw,T){e}_{\mathrm{ne}}({t}_{\mathrm{w}},T)ene​(tw​,T), of samples of large size Lb{L}_{\mathrm{b}}Lb​ are measured as a function of anneal time tw{t}_{\mathrm{w}}tw​ after a quench to temperatures TTT. The two observables are compared to the equilibrium spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{eq}}(L,T)$ and the equilibrium energy eeq(L,T){e}_{\mathrm{eq}}(L,T)eeq​(L,T), respectively, measured as functions of temperature TTT and system size LLL for a range of system sizes. For any time and temperature a nonequilibrium time-dependent length scale L∗(tw,T){L}^{*}({t}_{\mathrm{w}},T)L∗(tw​,T) can be defined by writing ${\ensuremath{\chi}}_{\mathrm{ne}}({t}_{\mathrm{w}},T)={\ensuremath{\chi}}_{\mathrm{eq}}({L}^{*},T)$ (or the equivalent expression for the energy). Our analysis shows that for all systems studied, an ``effective dynamical critical exponent'' parametrization L∗(tw,T)=A(T)t1∕z(T){L}^{*}({t}_{\mathrm{w}},T)=A(T){t}^{1∕z(T)}L∗(tw​,T)=A(T)t1∕z(T) fits the data well at each temperature within the whole temperature range studied, which extends from well above the critical temperature Tc{T}_{\mathrm{c}}Tc​ to near T=0T=0T=0 for dimension 2 or to well below Tc{T}_{\mathrm{c}}Tc​ for the other space dimensions studied. In addition, the data suggest that the dynamical exponent zzz varies smoothly when crossing the transition temperature.
By means of Monte Carlo simulations on the three-dimensional Ising spin-glass model, we have studied aging phenomena with various temperature(TTT)-change protocols. Particularly, a TTT-shift protocol, in which a system is first quenched to and aged for a period tw1t_{\rm w1}tw1​ at a temperature T1T_1T1​, and subsequently aged at a new temperature T2T_2T2​ is closely investigated. Most importantly, the mean size of domains, which is extracted from the replica-overlap function, was found to grow monotonically without any appreciable decrease by the TTT-change. We also found the relaxation of energy density and spin auto-correlation function during the TTT-shift process can be explained within the following picture: the dynamics finally crossovers to an isothermal aging of T2T_2T2​ at around t2≃tw1(eff)t_2 \simeq t_{\rm w1}^{\rm (eff)}t2​≃tw1(eff)​ after the TTT-change, where tw1(eff)t_{\rm w1}^{\rm (eff)}tw1(eff)​ is an effective waiting time related with tw1t_{\rm w1}tw1​, but in the transient regime t2<tw1(eff)t_2 \lt t_{\rm w1}^{\rm (eff)}t2​<tw1(eff)​, adjustment of the population of thermally active droplets from that at T1T_1T1​ to T2T_2T2​ takes place very slowly. Implications of the results on aging phenomena in other TTT-change protocols such as TTT-cycling and continuous TTT-change with an intermittent stop observed by simulations as well as experiments are discussed.