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arXiv:cond-mat/0506199 · Physical Review E

Reply to “Comment on ‘Scaling of the linear response in simple aging systems without disorder’ ”

Malte Henkel, Michel Pleimling

Published Aug 29, 2005• 7 citations
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Abstract

The value of the nonequilibrium exponent $a$ is measured in the two-dimensional Ising model quenched to below criticality from the dynamical scaling of the zero-field-cooled and the intermediate susceptibility. Our results fully reconfirm the expected value $a=1∕2$ but are inconsistent with the value $a=1∕4$, advocated by Corberi, Lippiello, and Zannetti, Phys. Rev. E 72, 028103 (2005).

DOI
10.1103/physreve.72.028104

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We have recently shown that in nonequilibrium spin systems at criticality the limit ${X}^{\ensuremath{\infty}}$ of the fluctuation-dissipation ratio $X(t,{t}_{w})$ for $t⪢{t}_{w}⪢1$ can be measured using observables such as magnetization or energy [Phys. Rev. E 68, 016116 (2003)]. Pleimling argues in a Comment [preceding paper, Phys. Rev. 70, 018101 (2004)] on our paper that for such observables correlation and response functions are dominated by one-time quantities dependent only on $t$, and are therefore not suitable for a determination of ${X}^{\ensuremath{\infty}}$. Using standard scaling forms of correlation and response functions, as used by Pleimling, we show that our data do have a genuine two-time dependence and allow $X(t,{t}_{w})$ and ${X}^{\ensuremath{\infty}}$ to be measured, so that Pleimling's criticisms are easily refuted. We also compare with predictions from renormalization-group calculations, which are consistent with our numerical observation of a fluctuation-dissipation plot for the magnetization that is very close to a straight line. A key point remains that coherent observables make measurements of ${X}^{\ensuremath{\infty}}$ easier than the traditionally used incoherent ones, producing fluctuation-dissipation plots whose slope is close to ${X}^{\ensuremath{\infty}}$ over a much larger range.

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In order to check on a recent suggestion that local scale invariance [M. Henkel, Phys. Rev. Lett. 87, 265701 (2001)] might hold when the dynamics is of Gaussian nature, we have carried out the measurement of the response function in the kinetic Ising model with Glauber dynamics quenched to T(C) in d=4, where Gaussian behavior is expected to apply, and in the two other cases of the d=2 model quenched to T(C) and to below T(C), where instead deviations from Gaussian behavior are expected to appear. We find that in the d=4 case there is an excellent agreement between the numerical data, the local scale invariance prediction and the analytical Gaussian approximation. No logarithmic corrections are numerically detected. Conversely, in the d=2 cases, both in the quench to T(C) and to below T(C), sizable deviations of the local scale invariance behavior from the numerical data are observed. These results do support the idea that local scale invariance might miss to capture the non-Gaussian features of the dynamics. The considerable precision needed for the comparison has been achieved through the use of a fast new algorithm for the measurement of the response function without applying the external field. From these high quality data we obtain a=0.27+/-0.002 for the scaling exponent of the response function in the d=2 Ising model quenched to below T(C), in agreement with previous results.

arXiv:cond-mat/0604321•Physical Review E•Oct 17, 2006•24 citations
The value of the nonequilibrium exponent aaa is measured in the two-dimensional Ising model quenched to below criticality from the dynamical scaling of the zero-field-cooled and the intermediate susceptibility. Our results fully reconfirm the expected value a=1∕2a=1∕2a=1∕2 but are inconsistent with the value a=1∕4a=1∕4a=1∕4, advocated by Corberi, Lippiello, and Zannetti, Phys. Rev. E 72, 028103 (2005).
We have recently shown that in nonequilibrium spin systems at criticality the limit ${X}^{\ensuremath{\infty}}$ of the fluctuation-dissipation ratio X(t,tw)X(t,{t}_{w})X(t,tw​) for t⪢tw⪢1t⪢{t}_{w}⪢1t⪢tw​⪢1 can be measured using observables such as magnetization or energy [Phys. Rev. E 68, 016116 (2003)]. Pleimling argues in a Comment [preceding paper, Phys. Rev. 70, 018101 (2004)] on our paper that for such observables correlation and response functions are dominated by one-time quantities dependent only on ttt, and are therefore not suitable for a determination of ${X}^{\ensuremath{\infty}}$. Using standard scaling forms of correlation and response functions, as used by Pleimling, we show that our data do have a genuine two-time dependence and allow X(t,tw)X(t,{t}_{w})X(t,tw​) and ${X}^{\ensuremath{\infty}}$ to be measured, so that Pleimling's criticisms are easily refuted. We also compare with predictions from renormalization-group calculations, which are consistent with our numerical observation of a fluctuation-dissipation plot for the magnetization that is very close to a straight line. A key point remains that coherent observables make measurements of ${X}^{\ensuremath{\infty}}$ easier than the traditionally used incoherent ones, producing fluctuation-dissipation plots whose slope is close to ${X}^{\ensuremath{\infty}}$ over a much larger range.
The authors use the method of 'damage spreading' to measure the time taken for the equilibrium damage (the magnetization) to disappear in finite systems (10<or=L<or=103) at the critical temperature of the infinite system. This time is found to scale as L z with an exponent z of 2.16+or-0.02. This value is compatible with the expected value of the dynamic exponent for the 2D Ising model, but they argue that this method is unable to measure fluctuations properly which they show are responsible for the higher values recently reported in the literature. They suggest that the slowest mode of relaxation has an exponent of z>or=2.27.
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.
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.
We study the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. We perform an exact analysis of the spherical model in any dimension D >2 and numerical simulations on the two-dimensional Ising model. These systems are shown to be ageing in the long-time regime, i.e. their two-time autocorrelation and response functions, and associated fluctuation-dissipation ratio, are non-trivial scaling functions of both time variables. We show in particular that, for 1<< s (waiting time) << t (observation time), the fluctuation-dissipation ratio possesses a non-trivial limit value X ∞ , which appears as a dimensionless amplitude ratio, and is therefore a novel universal characteristic of non-equilibrium critical dynamics. For the spherical model, we obtain X ∞ = 1-2/ D for 2< D <4 and X ∞ = ½ for D >4 (mean-field regime). For the two-dimensional Ising model we measure X ∞ ≈0.26±0.01.
In order to check on a recent suggestion that local scale invariance [M. Henkel, Phys. Rev. Lett. 87, 265701 (2001)] might hold when the dynamics is of Gaussian nature, we have carried out the measurement of the response function in the kinetic Ising model with Glauber dynamics quenched to T(C) in d=4, where Gaussian behavior is expected to apply, and in the two other cases of the d=2 model quenched to T(C) and to below T(C), where instead deviations from Gaussian behavior are expected to appear. We find that in the d=4 case there is an excellent agreement between the numerical data, the local scale invariance prediction and the analytical Gaussian approximation. No logarithmic corrections are numerically detected. Conversely, in the d=2 cases, both in the quench to T(C) and to below T(C), sizable deviations of the local scale invariance behavior from the numerical data are observed. These results do support the idea that local scale invariance might miss to capture the non-Gaussian features of the dynamics. The considerable precision needed for the comparison has been achieved through the use of a fast new algorithm for the measurement of the response function without applying the external field. From these high quality data we obtain a=0.27+/-0.002 for the scaling exponent of the response function in the d=2 Ising model quenched to below T(C), in agreement with previous results.