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Physical review. B, Condensed matter

Aging in two-dimensional Ising spin glasses

A. G. Schins, E. M. Dons, A. F. M. Arts, H. W. de Wijn, E. Vincent, L. Leylekian, J. Hammann

Published Dec 1, 1993• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.16524

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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} (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/\ensuremathψ{)}^{1/\mathrm{\ensuremath{\psi}}})1/\ensuremathψ, 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 ac susceptibility in the two-dimensional Ising spin glass Rb2{\mathrm{Rb}}_{2}Rb2​Cu0.67{\mathrm{Cu}}_{0.67}Cu0.67​Co0.33{\mathrm{Co}}_{0.33}Co0.33​F4{\mathrm{F}}_{4}F4​ and the two-dimensional random-exchange Ising ferromagnet Rb2{\mathrm{Rb}}_{2}Rb2​Cu0.89{\mathrm{Cu}}_{0.89}Cu0.89​Co0.11{\mathrm{Co}}_{0.11}Co0.11​F4{\mathrm{F}}_{4}F4​ 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.
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.
Aging phenomena of short-range Ising spin glass models have been investigated using Monte Carlo simulations. It is found that in the low-temperature spin-glass phase the mean domain size exhibits a crossover from a power-law growth associated with the critical fluctuation at the transition temperature to slower growth inherent in the low-temperature phase. The temperature dependence of the growth law of the domain size can be almost explained by this crossover. We also find that the spin-autocorrelation function in the quasi-equilibrium regime follows the expected scaling behavior from the droplet theory expressed in terms of the mean domain size and a characteristic length scale of droplet excitations. Spin glasses (SG) exhibit characteristic slow dynamics below the SG transition temperature. 1)-3) Recently such slow dynamics, in particular aging phenomena, has been much attractive both in experimental and theoretical studies. Several attempts have been made so far to explain the slow dynamics. There are mainly two distinct
Aging phenomena of short-range Ising spin glass models have been investigated using Monte Carlo simulations. It is found that in the low-temperature spin-glass phase the mean domain size exhibits a crossover from a power-law growth associated with the critical fluctuation at the transition temperature to slower growth inherent in the low-temperature phase. The temperature dependence of the growth law of the domain size can be almost explained by this crossover. We also find that the spin-autocorrelation function in the quasi-equilibrium regime follows the expected scaling behavior from the droplet theory expressed in terms of the mean domain size and a characteristic length scale of droplet excitations.
The low temperature dynamics of the three-dimensional Ising spin-glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function C(t, t w ) shows a crossover at the waiting (or aging) time t w from algebraic quasiequilibrium decay for times t<<t w to another, faster algebraic decay for t>>t w with an exponent similar to one for the remanent magnetization.
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.