Nonequilibrium dynamics of spin glasses: Examination of the ghost domain scenario

P. E. Jönsson, R. Mathieu, P. Nordblad, H. Yoshino, H. Aruga Katori, and A. Ito

Phys. Rev. B 70, 174402 – Published 3 November 2004

Abstract

Extensive experimental and numerical studies of the nonequilibrium dynamics of spin glasses subjected to temperature or bond perturbations have been performed to investigate chaos and memory effects in selected spin-glass systems. Temperature shift and cycling experiments were performed on the strongly anisotropic Ising-like system ${Fe}_{0.5} {Mn}_{0.5} {TiO}_{3}$ and the weakly anisotropic Heisenberg-like system Ag(11 at% Mn), while bond shift and cycling simulations were carried out on a four-dimensional Ising Edwards-Anderson spin glass. These spin-glass systems display qualitatively the same characteristic features, and the observed memory phenomena are found to be consistent with predictions from the ghost-domain scenario of the droplet scaling model.

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Received 25 July 2003

DOI:https://doi.org/10.1103/PhysRevB.70.174402

Authors & Affiliations

P. E. Jönsson^*, R. Mathieu^†, and P. Nordblad

Department of Materials Science, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden

H. Yoshino

Department of Earth and Space Science, Faculty of Science, Osaka University, Toyonaka, 560-0043 Osaka, Japan

H. Aruga Katori and A. Ito

RIKEN, Hirosawa 2-1, Wako, Saitama, 351-0198, Japan

^*Present address: RIKEN, Hirosawa 2-1, Wako, Saitama 351-0198, Japan.
^†Present address: ERATO-SSS/JST, Cryogenic Center, University of Tokyo, 2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan.

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Vol. 70, Iss. 17 — 1 November 2004

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Figure 1
A schematic picture of the configuration of the local overlap $O_{i}^{A B}$ between the equilibrium state $Γ_{A}$ and $Γ_{B}$ . The black and white represents $\pm 1$ values of the local overlap. As a first approximation, the configuration of the local overlap can be described as a collection of “blocks” of linear size of the overlap length $L_{Δ T}$ within which the sign of the overlap is biased to positive (black) or negative (white). However, there are minorities within the blocks that have the opposite sign of those of the majorities.Reuse & Permissions
Figure 2
(Color online) An illustration of the evolution of ghost domains in a “perturbation-healing” protocol at large length scales. The above pictures are generated by an artificial simulation: Monte Carlo simulation is performed on a set of Ising spins on a square lattice by cycling the working Hamiltonian given by Mattis models whose ground states are exactly given by $σ^{A}$ and $σ^{B}$ (see Ref. 51 for the details). The different colors represent the sign $+$ and $-$ of the projections ${\tilde{s}}_{i}^{A} (t)$ (top figures) and ${\tilde{s}}_{i}^{B} (t)$ (bottom figures) at the end of initial aging state of duration $t_{w}$ , at the end of the perturbation stage of duration $τ_{p}$ and after time $τ_{h}$ in the healing stage. Here the backbone spin configuration of the target equilibrium states $σ^{A}$ and $σ^{B}$ are completely unrelated random configurations so that the local overlap $O_{i}^{A B} = {\tilde{s}}_{i}^{A} (t) {\tilde{s}}_{i}^{B} (t) = σ^{A} σ^{B}$ [see Eq. (8)] is a completely random configuration: the overlap length is just 1 in the unit lattice. The remanent bias at the end of the perturbation stage is found as (left) $ρ_{rem} \sim 0.15$ and (right) $ρ_{rem} \sim 0.04$ . In the former the human eyes just barely distinguish the ghost domains, whereas in the latter the domains have become indistinguishable. In both cases, the strength of the bias increases up to $O (1)$ in the healing stage: (left) $ρ_{rec} \sim 0.59$ and (right) $ρ_{rec} \sim 0.58$ .Reuse & Permissions
Figure 3
(Color online) A schematic illustration of the temperature profiles of the effective domain size around the working temperatures $T_{A}$ and $T_{B}$ . The solid and dotted lines represent the effective domain size $L_{eff} (t)$ at logarithmically separated times $t$ during isothermal aging at $T_{A}$ and $T_{B}$ , respectively.Reuse & Permissions
Figure 4
(Color online) The ac susceptibility versus time after a temperature quench. A temperature cycling from 30 K to 28 K and back to 30 K is shown as pluses. The solid lines are the isothermal aging at 28 K and 30 K. After the negative $T$ shift from 30 K to 28 K, the system is completely rejuvenated. Still, after the $T$ cycling it keeps a memory of the first aging at 30 K. $f = 510 mHz$ .Reuse & Permissions
Figure 5
(Color online) Schematic representation of a twin $T$ -shift experiment. In the experiment represented using the thick line $(T_{i}, T_{m}) = (T_{1}, T_{2})$ , while $(T_{i}, T_{m}) = (T_{2}, T_{1})$ in the experiment represented by the thin line. In ac experiments, $χ (t)$ is also monitored during the halt at $T_{i}$ . In order to ensure thermal stability at $T_{m}$ , a short waiting time $t_{w s} = 3 s$ is needed before applying the magnetic field and recording the ZFC relaxation (Ref. 64).Reuse & Permissions
Figure 6
(Color online) Relaxation rate versus time after positive and negative twin $T$ -shift measurements. Pluses are positive $T$ shifts $(T_{1}, T_{2})$ , while squares are negative $T$ shifts $(T_{2}, T_{1})$ . (a) Ag(11 at% Mn): $T_{1} = 30 - ∣ Δ T ∣ K$ and $T_{2} = 30 K$ . (b) ${Fe}_{0.5} {Mn}_{0.5} {TiO}_{3}$ : $T_{1} = 19 - ∣ Δ T ∣ K$ and $T_{2} = 19 K$ . Note that for each $Δ t$ the peak position $t_{\max}$ increases with $t_{w}$ and the maximum in $S (t)$ broadens, which suggests that these data reflect the weakly perturbed regime of the temperature-chaos effect. (See Fig. 11 for the corresponding data of bond shifts in the 4D EA model.)Reuse & Permissions
Figure 7
(Color online) Relaxation rate versus time after aging the Ag(11 at% Mn) sample at (a) $T_{i} = 29.95$ ,29.9,29.8,29.7,29.6,29.5,29.4 K (b) $T_{i} = 30.1$ ,30.2,30.3,30.4,30.5,30.6,30.7 K using $t_{w} = 1000 s$ and $T_{m} = 30 K$ . Dotted lines are ordinary isothermal aging curves with $t_{w} = 3$ and 1000 s. [See Fig. 3 in Ref. 67 for the case of a Cu(Mn) spin glass.]Reuse & Permissions
Figure 8
(Color online) $χ^{″}$ versus time measured for positive and negative $T$ -shift experiments. (a) The Ag(11 at% Mn) sample is aged at $T_{i} = 29.95$ ,29.9,29.8,29.7,29.6 K for $t_{w} = 10 000 s$ before changing the temperature to $T_{m} = 30 K$ . (b) It is aged at $T_{i} = 30.1$ ,30.2,30.3,30.4,30.5 K for $t_{w} = 1000 s$ before changing the temperature to $T_{m} = 30 K$ . (c) The ${Fe}_{0.5} {Mn}_{0.5} {TiO}_{3}$ sample is aged at $T_{i} = 18.8$ ,18.5,18.0,17.0,16.0 K for $t_{w} = 10 000 s$ before changing the temperature to $T_{m} = 19 K$ . (d) It is aged at $T_{i} = 19 K$ for $t_{w} = 1000 s$ before changing the temperature to $T_{m} = 18.8$ ,18.5,18.0,17.0,16.0 K. The thick lines are the (reference) isothermal aging curves starting from $t_{w} + t = 0$ and $t_{w} + t = t_{w}$ . The insets show $χ^{″} (T_{m})$ versus $t + t_{eff}$ with $t_{eff}$ determined from ZFC-relaxation experiments. $f = 510 mHz$ .Reuse & Permissions
Figure 9
(Color online) Relation between $t_{w}$ and $t_{eff}$ in twin experiments: $(T_{1}, T_{2})$ shown as pluses and $(T_{2}, T_{1})$ shown as squares. (a) For the Ag(11 at% Mn) sample, $T_{1} = (30 - ∣ Δ T ∣) K$ and $T_{2} = 30 K$ with $∣ Δ T ∣ = 0.05$ ,0.1,0.2,0.3,0.4,0.5 K $(0.015 T_{g})$ . The solid line indicates fully accumulative aging $L_{T_{1}} (t_{1}) = L_{T_{2}} (t_{2})$ , calculated assuming logarithmic domain growth [Eq. (A2)]. (b) for the ${Fe}_{0.5} {Mn}_{0.5} {TiO}_{3}$ sample, $T_{1} = (19 - ∣ Δ T ∣) K$ and $T_{2} = 19 K$ with $∣ Δ T ∣ = 0.2$ ,0.5,1.0,1.5 K $(0.064 T_{g})$ .Reuse & Permissions
Figure 10
(Color online) Scaling plot of $L_{eff}$ with $L_{Δ T} = L_{0} {(a ∣ Δ T ∣ ∕ T_{g})}^{- 1 ∕ ζ}$ with $a = 6$ and $1 ∕ ζ = 2.6$ . The fit is due to the scaling ansatz for the weakly perturbed regime [Eq. (11)] with $c = 0.4$ . The deviation from accumulative aging is shown in the inset, and the line is the fit to the scaling ansatz for the weakly perturbed regime. (See Fig. 12 for the corresponding scaling plot of the data of bond shifts in the 4D EA model.)Reuse & Permissions
Figure 11
(Color online) The relaxation rate $S (t) = - d C (t + t_{w}, t_{w}) ∕ \log (t)$ of the 4D EA model after different waiting times $t_{w}$ with bond shifts of various strength $p$ at temperature $T = 0.6 T_{g}$ .Reuse & Permissions
Figure 12
(Color online) Test of the scaling of $L_{eff}$ after bond shifts in the 4D EA model. The straight line in the main figure is the no-chaos limit. The fit is due to the scaling ansatz for the weakly perturbed regime [Eq. (9)] with $L_{Δ J} (p) = 0.1 {\sqrt{p}}^{- 1 ∕ ζ}$ with $ζ = 0.9$ . The dynamical length $L_{T} (t)$ is due to the data obtained in Refs. 63, 69. Thus, there are essentially zero fitting parameters. The correction part to the no-chaos limit is shown in the inset, where the straight line is the power law with exponent $2 ζ$ . See the corresponding scaling plot in the case of the temperature-shift experiments on the Ag(11 at% Mn) sample shown in Fig. 10.Reuse & Permissions
Figure 13
(Color online) Schematic representation of a one-step temperature-cycling experiment according to the protocol $T_{m} (t_{w_{1}}) \to T_{m} + Δ T (t_{w_{2}}) \to T_{m}$ . When returning to $T_{m}$ , a short waiting time, $t_{w s} = 3 s$ is needed before applying the magnetic field and recording the ZFC relaxation in order to ensure thermal stability. In ac experiments, $χ (t)$ is also recorded during the cycling.Reuse & Permissions
Figure 14
(Color online) $χ^{″} (t)$ versus time at $T_{m}$ for experiments with a negative $T$ cycling of $Δ T = - 2 K (T_{m} + Δ T = 28 K); t_{w_{1}} = 3000 s$ and $t_{w_{2}} = 30$ ,300,3000,30 000 s (bottom to top). The thick (black) lines represent the reference isothermal aging at $T_{m}$ ; one of these curves is shifted a quantity $t_{w_{1}}$ in time. The inset shows $[χ^{″} (t) - χ_{ref}^{″} (t)] ∕ χ_{ref}^{″} (t)$ for the transient part of the susceptibility after the $T$ cycling. $χ^{″} (t)$ during the all stages of the $T$ cycling are shown in Fig. 4 for $t_{w_{2}} = 3000 s . ω ∕ 2 π = 510 mHz$ .Reuse & Permissions
Figure 15
(Color online) ZFC-relaxation curves [ $M_{ZFC} (t)$ and $S (t)$ ] obtained after a negative $T$ cycling of $Δ T = - 1.5 K (T_{m} = 30 K; T_{m} + Δ T = 28.5 K)$ ; Type I: $t_{w_{1}} = 30$ ,300, 3000, and 30 000 s and $t_{w_{2}} = 3000 s$ ; Type II: $t_{w_{1}} = 3000 s$ and $t_{w_{2}} = 30$ ,300,3000,30 000 s. Isothermal aging recorded at $T_{m} = 30 K$ are added for comparison in dashed lines; Type I: $t_{w} = t_{w_{1}}$ , Type II: $t_{w} = 3 s$ , and $t_{w} = 3000 s$ . (See Fig. 18 for the corresponding data of the one-step bond cycling in the 4D EA model.)Reuse & Permissions
Figure 16
Schematic illustration of (a) positive $(T_{2} > T_{1})$ and (b) negative $(T_{2} < T_{1})$ two-step $T$ cycling.Reuse & Permissions
Figure 17
(Color online) ZFC-relaxation curves obtained after negative two-step $T$ cyclings. Two-step $T$ -cycling experiments $T_{m} (t_{w_{1}}) \to T_{1} (t_{w_{2}}) \to T_{2} (t_{w_{3}}) \to T_{m}$ with $T_{m} = 30 K$ and $t_{w_{1}} = t_{w_{2}} = t_{w_{3}} = 3000 s$ are shown. A negative $T$ shift in $T_{1} \to T_{2}$ is made with $(T_{1}, T_{2}) = (29 K, 27 K)$ (diamonds), and positive $T$ shifts in $T_{1} \to T_{2}$ are made with $(T_{1}, T_{2}) = (28 K, 29 K)$ (circles), $(T_{1}, T_{2}) = (27 K, 29 K)$ (pulses). The isothermal aging with $t_{w} = 3$ and 3000 s (dotted lines) at $T_{m} = 30 K$ and one-step $T$ -cycling experiments $T_{m} (t_{w_{1}}) \to T_{m} + Δ T (t_{w_{2}}) \to T_{m}$ with $T_{m} = 30 K, T_{m} + Δ T = 29 K, t_{w_{2}} = 3000$ and 30 000 s (dashed lines) are shown for comparison. (See Fig. 19 for the corresponding data of the two-step bond cycling in the 4D EA model.)Reuse & Permissions
Figure 18
(Color online) ZFC susceptibility and the corresponding autocorrelation function in the 4D EA model after bond cyclings. The filled squared are the data obtained after bond cyclings with Type I: $t_{w_{1}} = 10^{2}, 10^{3}, 10^{4}, 10^{5} MCS$ ( $χ_{ZFC}$ from top to bottom $C_{ZFC}$ from bottom to top), $t_{w_{2}} = 10$ , and $t_{w_{3}} = 10 (MCS)$ ; Type II: $t_{w_{1}} = 10^{4} MCS, t_{w_{2}} = 10$ , 40, 160, 640, and $10^{5} MCS$ ( $χ_{ZFC}$ from top to bottom $C_{ZFC}$ from bottom to top), and $t_{w_{3}} = 10 MCS$ . The solid lines are the reference isothermal aging data: (a) $χ_{0} (t, t_{w})$ and (b) $C_{0} (t, t_{w})$ with $t_{w} = t_{w_{3}}$ and the dotted lines those with $t_{w} = t_{w_{1}} + t_{w_{3}}$ . (See Fig. 15 for the corresponding temperature-cycling experiment.)Reuse & Permissions
Figure 19
(Color online) The relaxation after two-step bond cycling $J^{A} \to J^{B} \to J^{C} \to J^{A}$ . The target equilibrium state is cycled as $J^{A} (10^{4} MCS) \to J^{B} (t_{w_{2}} MCS) \to J^{C} (t_{w_{3}} MCS) \to J^{A} (t MCS)$ . In the figures the dotted lines are the reference data of one-step bond-cycling $J^{A} (10^{4} MCS) \to J^{C} (t_{w_{2}} MCS) \to J^{A} (t MCS)$ with $t_{w_{2}} = 20$ ,40,80,160,320,640 MCS, which vary from bottom to top in (a) and from top to bottom in (b). (See Fig. 17 for the corresponding two-step temperature-cycling experiment.)Reuse & Permissions
Figure 20
(Color online) $χ^{″}$ versus temperature with temporary stops at (a) $T_{s_{1}} = 30 K$ and $T_{s_{2}} = 24 K$ and (b) $T_{s_{1}} = 20 K$ and $T_{s_{2}} = 16 K; t_{s_{1}} = t_{s_{2}} = 3000 s$ . These values of $T_{s}$ correspond to $T ∕ T_{f} = 0.9$ and 0.7 for both samples, where $T_{f}$ is the freezing temperature (Ref. 75). Filled symbols are used to represent the curves measured on cooling, and open symbols the curves measured on heating. The reference curves (without stops) are shown as lines; the arrows indicate the cooling respective of the heating curve. The insets show the difference plots for double-stop experiments (open symbols) and associated single-stop experiments (lines) measured on heating. $ω ∕ 2 π = 510 mHz$ and $v_{cool}^{eff} \approx v_{heat}^{eff} \sim 0.005$ .Reuse & Permissions
Figure 21
(Color online) ZFC susceptibility versus $T ∕ T_{f}$ for (a) the Ag(11 at% Mn) sample and (b) the ${Fe}_{0.5} {Mn}_{0.5} {TiO}_{3}$ sample. The ZFC susceptibility is measured after continuous cooling (filled symbols) and cooling with a temporary stop (open symbols) of 3000 s at $T_{s} ∕ T_{f} \sim 0.8$ ( $T_{s} = 27 K$ for Ag(11 at% Mn) and $T_{s} = 19 K$ for ${Fe}_{0.5} {Mn}_{0.5} {TiO}_{3}$ ). The insets show the corresponding difference plots (“stop curve”—reference curve), and additional results for 3000 s stops made at (a) $T_{s} = 24 – 31 K$ and (b) $T_{s} = 16 – 21 K$ . $v_{cool} \sim 0.05 K ∕ s$ and $v_{heat}^{eff} \sim 0.005$ .Reuse & Permissions
Figure 22
(Color online) Difference plots of ZFC susceptibility versus $T ∕ T_{f}$ corresponding to single- and double-stop experiments, with one (or two) temporary stops for 3000 s at (a) $T_{s_{1}} = 30 K$ and/or $T_{s_{2}} = 24 K$ and (b) $T_{s_{1}} = 21 K$ and/or $T_{s_{2}} = 17 K$ . The single-stop experiments are represented using filled markers, while open markers are used to represent the double-stop experiments. $v_{cool} \sim 0.05 K ∕ s$ and $v_{heat}^{eff} \sim 0.005$ .Reuse & Permissions
Figure 23
(Color online) A schematic illustration of the length scales explored in the Ag(11 at% Mn) sample by continuous cooling (and heating) with temporary halts at $T_{s_{1}} = 30 K$ and $T_{s_{2}} = 24 K$ for $t_{s_{1}} = t_{s_{2}} = 10^{4} s$ . The effective domain size grown during the temporary halts at $T_{s}$ for a time $t_{s}$ is given by $L_{eff} = L_{Δ T = ∣ T - T_{s} ∣} F (L_{T_{s}} (t_{s}) ∕ L_{Δ T = ∣ T - T_{s} ∣})$ [Eqs. (9, 10, 11, 12)]. It is bounded from above by the overlap length $L_{Δ T = ∣ T - T_{s} ∣}$ . The scaling function $F (x) = \tanh (x) [1 - c x^{2 ζ} ∕ \cosh (x)]$ interpolates between the fit to the weakly perturbed regime (shown in Fig. 10) and the strongly perturbed regime $\lim_{x \to \infty} F (x) = 1$ . Here all parameters for the growth law $L_{T} (t)$ [see Fig. 24] and the overlap length $L_{Δ T}$ are the same as in Sec. 4A4. The continuous cooling and heating yields the minimum effective domain size $L_{\min}$ at all temperatures. Here it is simply assumed to be $L_{\min} = L_{T} (t_{\min})$ with $t_{\min} = 100 s$ (Ref. 77).Reuse & Permissions
Figure 24
(Color online) The dynamical length scales relevant in our experiments and simulations. (a) A plot of the growth law [Eq. (A2)] with parameters determined for a Ag(11 at% Mn) sample in Ref. 29 from a scaling analysis of the relaxation of the ac susceptibility in isothermal aging in the range of 20–30 K. The growth law at the critical temperature $L (t) ∕ L_{0} = {(t ∕ τ_{m})}^{1 ∕ z} (z = 6.5)$ is also shown. [The possible $O (1)$ prefactor in front of Eq. (A2) and the critical growth law are not known.] The unit of time is chosen to be $τ_{m} \sim ℏ ∕ J \sim 10^{- 13} s$ . The experimental time and/or length scale is the area surrounded by the box. (b) The size of the domains of a 4D EA Ising spin-glass model determined directly in a previous numerical simulation (Ref. 69), monitoring the spatial correlation function between two real replicas undergoing isothermal aging. The dotted line is the power law for the critical regime $L (t) ∕ L_{0} = {(t ∕ τ_{m})}^{1 ∕ z}$ with $z = 4.98$ .Reuse & Permissions
Figure 25
(Color online) $S (t)$ of ZFC-relaxation measurements employing different wait times on (a) Ag(11 at% Mn) and (b) ${Fe}_{0.5} {Mn}_{0.5} {TiO}_{3}$ . The reduced temperature is 0.9 for both samples.Reuse & Permissions
Figure 26
(Color online) $S (t)$ curves obtained for Ag(11 at% Mn) after a rapid cooling $(v_{cool} \sim 0.05 K ∕ s)$ to $T_{m} = 30 K$ (solid line) and after a rapid cooling to 29 K, where the sample was kept for 3000 s and subsequently heated rapidly $(v_{heat} \sim 0.5 K ∕ s)$ to $T_{m} = 30 K$ (dashed line).Reuse & Permissions

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