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Physical review. B, Condensed matter

Reply to ‘‘Comment on ‘Temperature dependence of the response time of dilute metallic spin glasses’ ’’

R. Hoogerbeets, Wei-Li Luo, R. Orbach

Published May 1, 1987• 7 citations
Publisher

Abstract

In this response we reply to the criticism of Nordblad et al. on our measurements of the dynamic repsonse of dilute metallic spin glasses. We argue that we have, in our experiments on dilute metallic spin glasses, separated the thermalization dynamics from the dynamic response to a change in applied field. This is contrary to experiments with observation times that exceed the characteristic thermalization time. On the basis of this observation we show that our interpretation of our experimental results is completely consistent with a recent model calculation of the dynamic response of the infinite-range Ising spin-glass model.

DOI
10.1103/physrevb.35.7185

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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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Yucesoy, Katzgraber, and Machta Reply:

Burcu Yücesoy, Helmut G. Katzgraber, Jonathan L. Machta

In Phys. Rev. Lett. 110, 219701 (2013) [arXiv:1211.0843] Billoire et al. criticize the conclusions of our Letter [Phys. Rev. Lett. 109, 177204 (2012), arxiv:1206.0783]. They argue that considering the Edwards-Anderson and Sherrington-Kirkpatrick models at the same temperature is inappropriate and propose an interpretation based on the replica symmetry breaking theory. Here we show that the theory presented in the Comment does not explain our data on the Edwards-Anderson spin glass and we stand by our assertion that the low-temperature behavior of the Edwards-Anderson spin glass model does not appear to be mean-field like.

arXiv:1304.5210•Physical Review Letters•May 24, 2013•12 citations
In this response we reply to the criticism of Nordblad et al. on our measurements of the dynamic repsonse of dilute metallic spin glasses. We argue that we have, in our experiments on dilute metallic spin glasses, separated the thermalization dynamics from the dynamic response to a change in applied field. This is contrary to experiments with observation times that exceed the characteristic thermalization time. On the basis of this observation we show that our interpretation of our experimental results is completely consistent with a recent model calculation of the dynamic response of the infinite-range Ising spin-glass model.
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.
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.
The mean field theory discussed in the preceding chapters revealed a surprisingly rich structure and in particular a very complex ordered phase with many ‘pure’ states. However, we know from periodic systems that the mean field theory does not always give the right answer. In this and the next two chapters, we will discuss alternative approaches to the study of spin glasses with short-range interactions. The most important one will be the renormalization group which led to a deep understanding of phase transitions (and in particular critical behaviour) of nonrandom systems and which has also turned out to be extremely useful for the study of spin glasses. Together with Monte Carlo simulations and experimental data, this will give us a fairly complete picture, at least of static spin glass properties below T f . Critical behaviour near T f and scaling arguments for T < T f have also been considered for the spin glass dynamics but here the situation seems to be less clear. Depending on the system, the remanent magnetization decays with quite different decay laws (see Chapter 1), which indicates that not all dynamic processes are universal. However, all spin glasses have a huge range of characteristic relaxation times, ranging from 10 −13 s (the Korringa relaxation in a metallic spin glass) to 10 −6 – 10 −8 s. This long-time limit is not inherent to the system but is simply determined by the patience of an experimentalist (or the average time of a student's thesis).
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.
Reply to Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional ±J\pm J±J Ising model'
In Phys. Rev. Lett. 110, 219701 (2013) [arXiv:1211.0843] Billoire et al. criticize the conclusions of our Letter [Phys. Rev. Lett. 109, 177204 (2012), arxiv:1206.0783]. They argue that considering the Edwards-Anderson and Sherrington-Kirkpatrick models at the same temperature is inappropriate and propose an interpretation based on the replica symmetry breaking theory. Here we show that the theory presented in the Comment does not explain our data on the Edwards-Anderson spin glass and we stand by our assertion that the low-temperature behavior of the Edwards-Anderson spin glass model does not appear to be mean-field like.