Abstract
Experiments featuring nonequilibrium glassy dynamics under temperature changes still await interpretation. There is a widespread feeling that temperature chaos (an extreme sensitivity of the glass to temperature changes) should play a major role but, up to now, this phenomenon has been investigated solely under equilibrium conditions. In fact, the very existence of a chaotic effect in the nonequilibrium dynamics is yet to be established. In this article, we tackle this problem through a large simulation of the 3D EdwardsAnderson model, carried out on the Janus II supercomputer. We find a dynamic effect that closely parallels equilibrium temperature chaos. This dynamic temperaturechaos effect is spatially heterogeneous to a large degree and turns out to be controlled by the spinglass coherence length ξ. Indeed, an emerging lengthscale ξ^{*} rules the crossover from weak (at ξ ≪ ξ^{*}) to strong chaos (ξ ≫ ξ^{*}). Extrapolations of ξ^{*} to relevant experimental conditions are provided.
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Introduction
An important lesson taught by spin glasses^{1} regards the fragility of the glassy phase in response to perturbations such as changes in temperature—temperature chaos (TC)^{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}—in the couplings^{6,7,13,14} or in the external magnetic field^{5,20,21}. In particular, it is somewhat controversial^{22,23,24,25,26,27} whether or not TC is the physical mechanism underlying the spectacular rejuvenation and memory effects found in spin glasses^{28,29,30,31} and several other materials^{32,33,34,35,36}. Indeed, a major obstacle in the analysis of these nonequilibrium experiments is that TC is a theoretical notion which is solely defined in an equilibrium context.
Specifically, TC means that the spin configurations that are typical from the Boltzmann weight at temperature T_{1} are very atypical at temperature T_{2} (no matter how close the two temperatures T_{1} and T_{2} are).
This equilibrum notion of TC has turned out to be remarkably elusive, even in the context of MeanField models (i.e., models that can be solved exactly in the MeanField approximation). Indeed, establishing the existence of TC for the SherringtonKirkpatrick model has been a real tour de force^{12}. Although SherringtonKirkpatrick’s model is the MeanField model of more direct relevance for this work, let us recall for completeness that TC has been investigated as well in other MeanField systems named pspin models. In these models, groups of p ≥ 3 spins interact (instead, p = 2 for SherringtonKirkpatrick). Surprisingly enough, one finds different behaviors. On the one hand, we have a recent mathematical proof of the absence of TC in the homogeneous spherical pspin model^{37}, in agreement with a previous claim based on physical arguments^{38}. On the other hand, TC should be expected when one mixes several values of p^{39}, as confirmed by a quite recent mathematical analysis^{40,41,42,43}. Unfortunately, the mathematically rigorous analysis of TC in offequilibrium dynamics seems out of reach for now, even in the MeanField context.
In order to obtain experimentally relevant results, one needs to go beyond the MeanField approximation and study shortrange spin glasses, represented by the EdwardsAnderson model^{44,45}. In this case, analytical investigations are even more difficult, but the equilibrium notion of TC that we have outlined above has been studied through numerical simulations. Yet, these equilibrium simulations have been limited to small system sizes by the severe dynamic slowing down^{6,7,8,11,13,14,16,17,18,19}.
Here we tackle the problem from a different approach by showing that a nonequilibrium TC effect is indeed present in the dynamics of a large spinglass sample in three spatial dimensions (our simulations of the EdwardsAnderson model are carried out on the Janus II custombuilt supercomputer^{46}). In a reincarnation of the staticsdynamics equivalence principle^{47,48,49,50}, just as equilibrium TC is ruled by the system size, dynamic TC is found to be governed by the timegrowing spinglass coherence length ξ(t_{w}), where the waiting time t_{w} is the time elapsed since the system was suddenly quenched from some very high temperature to the working temperature T. Below the critical temperature, T < T_{c}, the spin glass is perennially out of equilibrium as evinced by the neverending (and sluggish) growth of glassy magnetic domains of size ξ(t_{w}), see refs. ^{51,52} for instance. Now, the extreme sampletosample variations found in small equilibrated systems^{16,17,19,53,54,55} translate into a strong spatial heterogeneity of dynamic TC. Despite such strong fluctuations, our largescale simulations allow us to observe traces of the effect even in averages over the whole system. In close analogy with equilibrium studies^{16}, however, dynamic TC can only be fully understood through a statistical analysis of the spatial heterogeneity. A crossover length ξ^{*} emerges such that TC becomes sizeable only when ξ(t_{w}) > ξ^{*}. We find that ξ^{*} diverges when the two observation temperatures T_{1} and T_{2} approach. The analysis of this divergence reveals that ξ^{*} is the nonequilibrium partner of the equilibrium chaotic length^{3,56}. The large values of ξ(t_{w}) that we reach with Janus II allow us to perform mild extrapolations to reach the most recent experimental regime^{57}.
In equilibrium, sampleaveraged signals of TC become more visible when the size of the system increases^{16}. Analogously, offequilibrium a weak chaotic effect grows with ξ(t_{w}) when the whole system is considered on average. Hoping that studying spatial heterogeneities will help us to unveil dynamic TC, we shall consider spatial regions of spherical shape and linear size ~ ξ(t_{w}), chosen randomly within a very large spin glass. Staticsdynamics equivalence suggests regarding these spheres as the nonequilibrium analog of small equilibrated samples of linear size ~ ξ(t_{w}). The analogy with equilibrium studies^{16,17,19} suggests that a small fraction of our spheres will display strong TC. The important question will be how this rareevent phenomenon evolves as ξ(t_{w}) grows (in equilibrium, the fraction of samples not displaying TC is expected to diminish exponentially with the number of spins contained in the sample^{12,15}).
Results and discussion
Model
We simulate the standard EdwardsAnderson model in a threedimensional cubic lattice of linear size L = 160 and periodic boundary conditions. In each lattice node x, we place an Ising spin (S_{x} = ± 1). Lattice nearestneighbors spins interact through the Hamiltonian H = − ∑_{〈x, y〉}J_{xy}S_{x}S_{y} . The couplings J_{xy} are independent and identically distributed random variables (J_{xy} = ± 1 with 1/2 probability), fixed when the simulation starts (quenched disorder). This model exhibits a spinglass transition at temperature T_{c} = 1.1019(29)^{58}. We refer to each realization of the couplings as a sample. Statistically independent simulations of a given sample are named replicas. We have considerably extended the simulation of Baity et al.^{51}, by simulating N_{Rep} = 512 replicas (rather than 256) of the same N_{S} = 16 samples considered in^{51}, in the temperature range 0.625 ≤ T ≤ 1.1.
We simulate the nonequilibrium dynamics with a Metropolis algorithm. In this way, one picosecond of physical time roughly corresponds to a fulllattice Metropolis sweep. At the initial time t_{w} = 0 the spin configuration is fully random (i.e., we quench from infinite temperature). The subsequent growth of spinglass domains is characterized by the spinglass coherence length ξ(t_{w}). Specifically, we use the ξ_{1,2} integral estimators, see refs. ^{48,51,59,60} for details [the main steps in the computation of ξ_{1,2} are also sketched in Eqs. (8–10)), where one should set T_{1} = T_{2} = T].
Finally, let us briefly comment on our choices for N_{Rep} and N_{S}. A detailed analysis^{51,61} shows that, for a given total numerical effort N_{S} × N_{Rep}, errors in ξ are minimized if N_{Rep} ≫ N_{S}. Furthermore, Supplementary Note 1 shows that having N_{Rep} ≫ 1 is crucial as well for the main quantities considered in this work (see definitions below). Therefore, given our finite computational resources, we have chosen to limit ourselves to N_{S} = 16. This small number of samples is partly compensated by the fact that we are working close to the experimental regime L ≫ ξ [we remark that N_{S} = 1 in typical experiments: indeed, staticsdynamics equivalence suggests that the number of statistically independent events is proportional to N_{S}(L/ξ)^{3}].
The local chaotic parameter
We shall compare the spin textures from temperature T_{1} and waiting time t_{w1} with those from temperature T_{2} and waiting time t_{w2} (we consider T_{1} ≤ T_{2} ≤ T_{c}). A fair comparison requires that the two configurations be ordered at the same lengthscale, which we ensure by imposing the condition
A first investigation of TC is shown in Fig. 1. The overlap, computed over the whole sample, of two systems satisfying condition Eq. (1) is used to search for a coarsegrained chaotic effect. The resulting signal is measurable but weak. Instead, as explained in the introduction, spin configurations should be compared locally. Specifically, we consider spherical regions. We start by choosing N_{sph} = 8000 centers for the spheres on each sample. The spheres’ centers are chosen randomly, with uniform probability, on the dual lattice which, in a cubic lattice with periodic boundary conditions, is another cubic lattice of the same size, also periodic boundary condition. The nodes of the dual lattice are the centers of the elementary cells of the original lattice. The radii of the spheres are varied, but their centers are held fixed. Let B_{s,r} be the sth ball of radius r. Our basic observable is the overlap between replica σ (at temperature T_{1}), and replica τ ≠ σ (at temperature T_{2}):
where N_{r} is the number of spins in the ball, and t_{w1} and t_{w2} are chosen according to Eq. (1). Averages over thermal histories, indicated by 〈…〉_{T}, are computed by averaging over σ and τ.
Next, we generalize the socalled chaotic parameter^{6,16,17,20} as
The extremal values of the chaotic parameter have a simple interpretation: \({X}_{{T}_{1},{T}_{2}}^{s,r}=1\) corresponds with a situation in which spin configurations in the ball B_{s,r}, at temperatures T_{1} and T_{2}, are completely indistinguishable (absence of chaos) while \({X}_{{T}_{1},{T}_{2}}^{s,r}=0\) corresponds to completely different configurations (strong TC). A representative example our results is shown in Fig. 2.
Our main focus will be on the distribution function \(F(X,{T}_{1},{T}_{2},\xi ,r)=\,{\text{Probability}}\,[{X}_{{T}_{1},{T}_{2}}^{s,r}(\xi ) \,< \,X]\) and on its inverse X(F, T_{1}, T_{2}, ξ, r).
The rareevent analysis
Representative examples of distribution functions F(X, T_{1}, T_{2}, ξ, r) are shown in Fig. 3. We see that, in close analogy with equilibrium systems^{16,17,19}, while most spheres exhibit a very weak TC (X > 0.9, say), there is a fraction of spheres displaying smaller X (stronger chaos). Note that the probability F of finding spheres with X smaller than any prefixed value increases when ξ grows.
In order to make the above finding quantitative, we consider the (inverse) distribution function X(F, T_{1}, T_{2}, ξ, r). We start by fixing (T_{1}, T_{2}), ξ and some small probability F, which leaves us with a function of only r. In order to obtain smoother interpolations for small radius, however, we have used \({N}_{r}^{1/3}\) instead of r as our independent variable, a technical detailed discussion can be found in Supplementary Note 3.
Figure 4 shows plots of 1 − X under these conditions, which exhibit welldefined peaks (see further information about the fitting function to the peaks in the Supplementary Note 2). Now, to a first approximation we can characterize any peak by its position, height and width. Fortunately, these three parameters turn out to describe the scaling with ξ of the full 1 − X curve, see Supplementary Note 4.
The physical interpretation of the peak’s parameters is clear. The peak’s height represents the strength of dynamic TC (the taller the peak, the larger the chaos). The peak’s position indicates the optimal lengthscale for the study of TC, given the probability F, ξ and the temperatures T_{1}, T_{2}. The peak’s width indicates how critical it is to spot this optimal lengthscale (the wider the peak, the less critical the choice). Perhaps unsurprisingly, the peak’s position is found to scale linearly with ξ, while the peak’s width scales as ξ^{β}, with β slightly larger than one, see Supplementary Note 5 for further details. We shall focus here on the temperature and ξ dependence of the peak’s height (i.e., the strength of chaos), which has a richer behavior.
The ξ dependence of the peak’s height (for a given probability F and temperatures T_{1} and T_{2}) turns out to be reasonably well described by the following ansatz:
This formula describes a crossover phenomenon, ruled by a characteristic length ξ^{*}. For ξ ≪ ξ^{*} the peak’s height grows with ξ as a power law, while for ξ ≫ ξ^{*} the strongchaos limit [i.e., (1 − X) → 1] is approached. However, some consistency requirements should be met before taking the crossover length ξ^{*} seriously. Not only should the fit to Eq. (4) be of acceptable statistical quality (the fit parameters are the characteristic lengthscale ξ^{*} and the exponent α). One would also wish exponent α to be independent of the temperatures T_{1} and T_{2} and of the chosen probability F.
We find fair fits to Eq. (4), see Table 1. In all cases, exponent α turns out to be compatible with 2.1 at the twoσ level [except for the (F = 0.01, T_{1} = 0.625, T_{2} = 0.8) fit]. Under these conditions, we can interpret ξ^{*} as a characteristic length indicating the crossover from weak to strong TC, at the probability level indicated by F. Furthermore, the relatively large value of exponent α indicates that this crossover is sharp.
The trends for the crossover length ξ^{*} in Table 1 are very clear: ξ^{*} grows upon increasing F or upon decreasing T_{2} − T_{1}. Identifying ξ^{*} as the nonequilibrium partner of the equilibrium chaotic length ℓ_{c}(T_{1}, T_{2})^{3,56} will allow us to be more quantitative (indeed, the two lengthscales indicate the crossover between weak chaos and strong chaos). Now, the equilibrium ℓ_{c}(T_{1}, T_{2}) has been found to scale for the 3D Ising spin glass as
with ζ ≈ 1.07^{14} or ζ ≈ 1.07(5)^{16}. These considerations suggest the following ansatz for the nonequilibrium crossover length
where B(F, T_{1}) is an amplitude. We have tested Eq. (6) by computing a joint fit for four (T_{1}, F) pairs as functions of T_{2} − T_{1}, allowing each curve to have its own amplitude but enforcing a common ζ_{NE} (see Fig. 5). The resulting χ^{2}/d.o.f. = 7.55/7 validates our ansatz, with an exponent ζ_{NE} = 1.19(2) fairly close to the equilibrium result ζ = 1.07(5)^{16}. This agreement strongly supports our physical interpretation of the crossover length. We, furthermore, find that B is only weakly dependent on T_{1}. Nevertheless, the reader should be warned that it has been suggested^{16} that the equilibrium exponent ζ may be different in the weak and strongchaos regimes.
Conclusions
We have shown that the concept of temperature chaos can be meaningfully extended to the nonequilibrium dynamics of a large spin glass. This is, precisely, the framework for rejuvenation and memory experiments^{28,29,30,31}, as well as other more chaosoriented experimental work^{57}. Therefore, our precise characterization of dynamical temperature chaos paves the way for the interpretation of these and forthcoming experiments. Our simulation of spinglass dynamics doubles the numerical effort in^{51} and has been carried out on the JanusII specialpurpose supercomputer.
The key quantity governing dynamic temperature chaos is the timedependent spinglass coherence length ξ(t_{w}). The very strong spatial heterogeneity of this phenomenon is quantified through a distribution function F. This probability can be thought of as the fraction of the sample that shows a chaotic response to a given degree. When comparing temperatures T_{1} and T_{2}, the degree of chaoticity is governed by a lengthscale ξ^{*}(F, T_{1}, T_{2}). While chaos is very weak if ξ(t_{w}) ≪ ξ^{*}(F, T_{1}, T_{2}), it quickly becomes strong as ξ(t_{w}) approaches ξ^{*}(F, T_{1}, T_{2}). We find that, when T_{1} approaches T_{2}, ξ^{*}(F, T_{1}, T_{2}) appears to diverge with the same critical exponent that it is found for the equilibrium chaotic length^{16}.
Although we have considered in this work fairly small values of the chaotic system fraction F, a simple extrapolation, linear in \(\mathrm{log}\,F\), predicts ξ^{*} ≈ 60 for F = 0.1 at T_{1} = 0.7 and T_{2} = 0.8 (our closest pair of temperatures in Table 1). A spinglass coherence length well above 60a_{0} is experimentally reachable nowadays^{52,57,62,63} (a_{0} is the typical spacing between spins), which makes our dynamic temperature chaos significant. Indeed, while completing this manuscript, a closely related experimental study^{57} reported a value for exponent ζ_{NE} in fairly good agreement with our result of ζ_{NE} = 1.19(2) in Fig. 5.
Let us conclude by commenting on possible venues for future research. Clearly, it will be important to understand in detail how dynamic temperature chaos manifests itself in nonequilibrium experiments. Simple protocols (in which temperature sharply drops from T_{2} to T_{1}, see, e.g., Zhai et. al.^{57}) seems more accessible to a first analysis than memory and rejuvenation experiments^{28,29,30,31}. An important problem is that the correlation functions that are studied theoretically are not easily probed experimentally. Instead, experimentalists privilege the magnetization density (which is a spatial average over the whole sample). Therefore an important theoretical goal is to predict the behavior of the nonequilibrium timedependent magnetization upon a temperature drop. One may speculate that the Generalized FluctuationDissipation Relations^{64} might be the route connecting the correlation functions with the response to an externally applied magnetic field. Interestingly enough, these relations (that apply at fixed temperature) can be defined locally as well^{65}. The resulting spatial distribution function allows the reconstruction of the global response to the magnetic field. Extending this analysis to a temperature drop may turn out to be fruitful in the future.
Methods
All the observables involved in the computation of temperature chaos depend on a pair of replicas (σ, τ). The basic quantity is the overlap field
Usually, this pair of replicas are at the same temperature T. All the definitions are, however, straightforwardly extended to two temperatures. For instance, the fourpoint twotemperature spatial correlation function is
where […]_{J} denotes the average over the samples. Building on this function we can define our integral estimator for the coherence length^{60}:
and
As explained in the main text, times t_{w1} and t_{w2} are fixed through the condition expressed in Eq. (1), which ensures that we are comparing spin configurations that are ordered on the same length scale.
Since our t_{w} are on a discrete grid, we solve Eq. (1) for the global overlaps through a (bi)linear interpolation.
Data availability
The data contained in the figures of this paper, accompanied by the gnuplot script files that generate these figures, are publicly available at https://github.com/JanusCollaboration/caosdin.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We are grateful for discussions with R. Orbach and Q. Zhai. This work was partially supported by Ministerio de Economía, Industria y Competitividad (MINECO, Spain), Agencia Estatal de Investigación (AEI, Spain), and Fondo Europeo de Desarrollo Regional (FEDER, EU) through Grants No. FIS201676359P, No. PID2019103939RBI00, No. PGC2018094684BC21 and PGC2018094684BC22, by the Junta de Extremadura (Spain) and Fondo Europeo de Desarrollo Regional (FEDER, EU) through Grant No. GRU18079 and IB15013 and by the DGAFSE (Diputación General de Aragón – Fondo Social Europeo). This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant No. 694925LotglasSy). DY was supported by the Chan Zuckerberg Biohub and IGAP was supported by the Ministerio de Ciencia, Innovación y Universidades (MCIU, Spain) through FPU grant no. FPU18/02665. BS was supported by the Comunidad de Madrid and the Complutense University of Madrid (Spain) through the Atracción de Talento program (Ref. 2019T1/TIC12776).
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J.M.G.N. and D.N. contributed to Janus/Janus II simulation software. D.I., A.T. and R.T. contributed to Janus II design. M.B.J., E.C., A.C., L.A.F, J.M.G.N., I.G.A.P., A.G.G., D.I., A.M., A.M.S., I.P., S.P.G., S.F.S., A.T. and R.T. contributed to Janus II hardware and software development. L.A.F., V.M.M. and J.M.G. designed the research. J.M.G. analyzed the data. M.B.J., L.A.F., E.M., V.M.M., J.M.G., I.P., G.P., B.S., J.J.R.L., F.R.T. and D.Y. discussed the results. V.M.M., J.M.G., B.S. and D.Y. wrote the paper.
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BaityJesi, M., Calore, E., Cruz, A. et al. Temperature chaos is present in offequilibrium spinglass dynamics. Commun Phys 4, 74 (2021). https://doi.org/10.1038/s42005021005659
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DOI: https://doi.org/10.1038/s42005021005659
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