Non-equilibrium evolution of window overlaps in spin glasses
We investigate numerically the time dependence of “window” overlaps in a three-dimensional Ising spin glass below its transition temperature after a rapid quench. Using an efficient GPU implementation, we are able to study large systems up to lateral length and up to long times of sweeps. We find that the data scales according to the ratio of the window size to the non-equilibrium coherence length . We also show a substantial change in behavior if the system is run for long enough that it globally equilibrates, i.e. , where is the lattice size. This indicates that the local behavior of a spin glass depends on the spin configurations (and presumably also the bonds) far away. We compare with similar simulations for the Ising ferromagnet. Based on these results, we speculate on a connection between the non-equilibrium dynamics discussed here and averages computed theoretically using the “metastate”.
pacs:75.50.Lk, 75.40.Mg, 75.10.Nr
Spin glasses Binder and Young (1986); Mézard et al. (1987); Young (1998) below their transition temperature are not in equilibrium, except for very small sizes in some simulations. One therefore needs to be able to describe non-equilibrium behavior, and a lot of attention numerically Rieger (1993); Kisker et al. (1996); Marinari et al. (1996); Yoshino et al. (2002); Belletti et al (2009); Manssen and Hartmann (2014) has been focussed on the evolution of the system after a rapid quench to temperature below the transition temperature . Locally, spins establish correlations so one anticipates that they will be correlated up to some distance, the coherence length , which slowly increases with time. For distances longer than correlations will decay exponentially, while at shorter distances they will decay more slowly than that. Empirically one finds Rieger (1993); Kisker et al. (1996); Marinari et al. (1996); Yoshino et al. (2002); Belletti et al (2009); Manssen and Hartmann (2014) that the growth of is compatible with a small power of (although a logarithmic growth cannot be fully excluded using the available data), written as where, , a non-equilibrium dynamic exponent is found to depend on the ratio .
To understand the nature of the spin glass state one needs local probes, see e.g. Newman and Stein (2003) and references therein. A useful local probe is the distribution of the overlap of the spins in two copies of the system in a window of linear size . Equilibrium properties of window overlaps have been studied numerically before Marinari et al. (1998), but here we focus on their non-equilibrium behavior, which has not received much attention before apart from Ref. Marinari et al. (1996) which studied the non-equilibrium evolution of a dimensionless ratio of cumulant averages evaluated in windows of different size. In this paper we study the time dependence of the window overlap distribution , in a non-equilibrium situation. We find that the distribution scales as a function of the ratio of the window size to . The non-equilibrium window overlap distribution is very different from the global equilibrium overlap distribution in the mean field theory of Parisi Parisi (1979, 1980, 1983). In particular depends quite strongly on . However, if the system is run for a time long enough for the system to globally equilibrate, i.e. , then we find a change in the form of , which happens rapidly when viewed on a logarithmic time scale, such that then has a rather a weak dependence on and is quite similar to the value of Parisi’s global equilibrium overlap distribution Parisi (1983) . The strong change in behavior when indicates that local spin correlations are sensitive to spin orientations, and presumably also to the values of the interactions, at large (or at least intermediate) distances.
The theoretical description of spin glasses below is complicated. One approach developed in recent years is known as the “metastate” Newman and Stein (1996a, 1997a); Aizenman and Wehr (1990). In this paper we also speculate on a possible connection between non-equilibrium correlations following a quench, and averages computed according to the metastate.
The plan of this paper is as follows. In Sec. II we describe the metastate and a possible connection between quantities calculated from it and non-equilibrium averages following a quench. The model we simulate and the quantities we calculate are described in Sec. III. The results of the simulations are given in Sec. IV, together with corresponding results for a pure Ising ferromagnet, and our conclusions summarized in Sec. V.
Ii Averaging in spin glasses; the Metastate and Dynamics
In systems undergoing phase transitions it is desirable to know what are the various possible states to which the system can evolve below the transition temperature . A simple example is the Ising ferromagnet in zero magnetic field for which there is just a pair of states below , related by flipping all the spins, the “up”and “down” spin states. If the system is in one of these states then “connected” correlation functions vanish at large distances, e.g.
which is known as “clustering” of the correlation functions. The angular brackets denote a thermal average, here restricted to one of these states to capture the symmetry breaking. By contrast if we simply perform the Boltzmann sum we give equal weight to both of these states, the symmetry is not broken so , and hence the two terms in Eq. (1) do not cancel at large distances and the correlation functions do not have a clustering property. States which do not have a clustering property are called “mixed” states and those that do, like the “up” and “down” states of the ferromagnet, are called “pure” states.
To keep the description of the system as simple as possible it is desirable to use clustering (i.e. pure) states. In many cases this is easy because they are just the different states in which a global symmetry of the Hamiltonian is broken. However, in more complicated situations such as spin glasses, there can be pure states not related by any symmetry and so characterizing them can be quite difficult. Nonetheless, it is argued Fisher and Huse (1986, 1987a, 1987b, 1988); Newman and Stein (1992, 1996a, 1997a, 2003); Read (2014) to be important to describe spin glasses in terms of pure states rather than by computing the Boltzmann sum. The latter is done, for example, in the Parisi Parisi (1979, 1980, 1983) solution of the infinite-range Sherrington-Kirkpatrick (SK) Sherrington and Kirkpatrick (1975) model.
To define pure states in general, consider the situation in Fig. 1. The overall system is of very large size and has free or periodic boundary conditions. We compute the thermal average exactly, and determine the correlation functions in a much smaller window of size , somewhere in the bulk of the system. These correlations may have the clustering property, in which case the window is in a pure state, or they may not in which case it is in a mixed state. In fact, since we consider only zero field, states come in symmetry-related pairs, so the simplest situation would be a single pair of pure states.
But for a system like a spin glass, the correlations in the window could depend sensitively on the choice of interactions in distant regions of the of the system, perhaps even in a chaotic manner, an aspect first pointed out explicitly by Newman and Stein (NS) Newman and Stein (1992). To investigate this we divide the system of size into an inner region of size , larger than the window of size which is in the middle of it, and an outer region between and . We then change the bonds in the outer region and recompute the correlation functions in the window. Eventually we let all sizes tend to infinity with . It is possible that the state of the window is always the same as one changes the bonds in the outer region. However, it is also possible that the state changes, perhaps chaotically, as one changes the bonds in the outer region. Several possible situations have been discussed in detail:
For each set of bonds in the outer region one has only a single pair of pure states, and one finds the same pair for every set of outer bonds. This is called the “droplet model” the theory for which has been developed in the greatest detail by Fisher and Huse Fisher and Huse (1986, 1987a, 1987b, 1988).
For each set of bonds in the outer region one has only a single pair of pure states, but this pair varies chaotically as one changes the outer bonds. This is the “chaotic pairs” picture of NS Newman and Stein (1992).
For each set of bonds in the outer region one has a mixed state, and this mixes changes in a chaotic way as the outer bonds are changed. This is called the“replica symmetry breaking” (RSB) picture
1since it is the generalization to finite-range models of Parisi’s Parisi (1979, 1980, 1983) solution of the infinite-range SK model. The name arises because Parisi’s original solution used the replica method to average over the disorder.
In order to describe the states of a spin glass one needs to give a statistical description of the different states the window can be in as the bonds in the outer region are varied. NS Newman and Stein (1996a, 1997a) call this the metastate. The description that we give here is actually a little different from that of NS and is due to Aizenman and Wehr (AW) Aizenman and Wehr (1990). In NS’s approach there is no intermediate scale and one looks at the correlations in the window as the system size is grown leaving the bonds already present unchanged. It is expected Newman and Stein (1997b) that the two forms of the metastate are equivalent. In agreement with Read Read (2014) we find that it is easier to discuss the AW metastate.
The AW metastate average is therefore performed by first doing a thermal average for the whole system, denoted by , followed by an average over the bonds in the outer region, denoted by . Following Read Read (2014) we call this the metastate-averaged state (MAS). Hence, if and lie within the window, the spin glass correlation function of their spins in the MAS is given by
(note the location of the square). After this average is done one can also average over the bonds in the inner region, which we denote by . We will present data for the window overlap distribution for which averaging over the bonds in the inner region is, strictly, speaking, unnecessary since translation invariant MAS averages are self-averaging Newman and Stein (1996b, a, 1997a). However, in practice, this last average is done in simulations to improve statistics.
It is interesting to ask how the MAS average varies at large distance according to the three scenarios mentioned above:
In the droplet picture one finds always the same pair of thermodynamic states so presumably
where is the Edwards-Anderson order parameter, which is well defined if we add a small symmetry-breaking field to remove the degeneracy between the pair of pure states. Equation (3) then follows because of clustering of correlations in a single pure state, see Eq. (1). We should mention, though, that the approach to the constant value of is expected to be quite slow, a power-law rather than an exponential, and so, for the values of that one can simulate, one may be far from the constant value (David Huse, private communication).
In the chaotic pairs picture, correlations in the window alter, in sign as well as magnitude, as the outer bonds are varied. Hence, according to Read Read (2014), it is expected that tends to zero, presumably as a power law, which is commonly written as
for , which defines the exponent .
In the RSB picture, which also has many states, MAS averaged correlations are similarly expected to decay as the power law in Eq. (4). In fact has been calculated in mean field theory de Dominicis et al. (1998); De Dominicis and Giardina (2006); Marinari et al. (2000); Read (2014) (corresponding to ) assuming RSB, with the result .
A large spin glass system is not in thermal equilibrium below . Results from the Boltzmann sum do not, therefore, correspond to experimental observations which are inevitably in a non-equilibrium situation. Are MAS averages any better in this regard? It is tempting to think so for the following reason.
Imagine quenching the spin glass to below and observing correlations in a local window of size . Correlations will develop up to some coherence length which grows slowly with time. How does one expect the non-equilibrium correlation function
where denotes an average over all the bonds, to vary as a function of ? Let us assume that time is large enough that . We postulate that thermal fluctuations of the spins outside the window at a distance and greater, which are not equilibrated with respect to spins in the window, effectively generate a random noise to the spins in the window which plays a similar role to the random perturbation coming from changing the bonds in the outer region according to the AW metastate, see Fig. 1. Thus we suggest that is analogous to the intermediate scale , separating inside and outside regions, in the construction of the metastate. This is indicated in Fig 1. After this work was submitted it was brought to our attention that a similar picture of non-equilibrium dynamics following a quench was discussed earlier by White and Fisher White and Fisher (2006). They denote the state obtained after a quench as the “maturation metastate” and the distribution of states in the AW or NS picture as the “equilibrium metastate”. Here we speculate that these might be the same. We thank Nick Read for bringing this paper to our attention.
This analogy suggests that the decay of correlations determined from the metastate may be the same as the decay of correlations following a quench, on scales shorter than the coherence length. We note that NS have also discussed dynamics following a quench Newman and Stein (2008, 1999) from a rigorous point of view.
There have been many simulations which investigate the time dependence of correlations following a quench Rieger (1993); Kisker et al. (1996); Marinari et al. (1996); Yoshino et al. (2002); Belletti et al (2009); Manssen and Hartmann (2014). Interestingly these papers do see a power law decay of the correlation function in Eq. (5) for sufficiently long times that , i.e.
The exponent is found to be about in three dimensions Rieger (1993); Kisker et al. (1996); Marinari et al. (1996); Yoshino et al. (2002); Belletti et al (2009); Manssen and Hartmann (2014). Equation (6) is of the same form as Eq. (4) which is obtained from metastate calculations for the Edwards-Anderson model Marinari et al. (2000); Read (2014) in the mean field approximation, assuming the RSB picture. The droplet theory predicts a different result, namely Eq. (3), though, of course, the numerical data may not be at large enough length scales to be in the asymptotic scaling regime.
Iii The model and quantities to be calculated
We simulate the Edwards-Anderson Edwards and Anderson (1975) Ising spin glass model with Hamiltonian
where the spins take values and are on the sites of a simple cubic lattice with spins with periodic boundary conditions. The quenched interactions are between nearest neighbors and take values with equal probability. The latest determination of the transition temperature of this model is Baity-Jesi et al (2013). Here we work at a fixed temperature of . Most of the simulations are for system size , which can not be brought to equilibrium in available computer time, but we also perform some simulations at smaller sizes to investigate the change in behavior when the system reaches global equilibrium. The number of samples simulated for each size is shown in Table 1.
We run two copies of the system with the same bonds but different initial random spin configurations, which we quench to at time , and then let the system evolve. To perform long runs on large lattices we have implemented an efficient Monte Carlo code on GPUs, see Manssen and Hartmann (2014) for details. At a logarithmically increasing set of times we store the spin configurations from which we calculate the correlation function in Eq. (5) as a function of at different times.
We also compute the time-dependent window overlap distribution defined by
where , the window overlap between replicas “” and “”, is
in which the sum is over the sites in the window and, for ease of notation, we have suppressed an index on which would indicate that it also depends on time. To improve statistics we average over all non-overlapping windows of size . The number of these is where indicates rounding down to the nearest integer. In addition, we smooth the data by computing, for each discrete value of the overlap, say, an average of the distribution on neighboring -values weighted by a normalized kernel which falls to zero as increases Alvarez Banos et al (2010).
iv.1 Spin Glass
An example of our data for the window overlap distribution is shown in Fig. 2 for . One sees an evolution from a single peak structure at short times, presumably Gaussian, to a two-peaked structure at long times. For larger window sizes, the distribution evolves more slowly, as shown in the data for , the weight of the distribution at , for different sizes in Fig. 3.
We would like to perform a scaling collapse of the data in Fig. 3 to ascertain the dependence of on and . However, rather than scaling with respect to we find it better to scale with the dynamic coherence length . At long times, where , the time-dependent correlation function in Eq. (5) varies with an inverse power of as shown in Eq. (6), so a natural scaling ansatz is
The coherence length can be taken from a ratio of moments of Belletti et al (2009), e.g.
Note that this calculation of did not make any reference to a window. However, if we compute the second moment of the window overlap distribution, , we note first that it is just the average of over all sites and in the window since
Using Eq. (10) this can be written as
where we used the substitution in the next to last line.
If we divide by an arbitrary scale factor the distribution of is where because both distributions are normalized. If we take , the standard deviation of , then . But has standard deviation unity, and so, if the distribution is smooth and extends down to the origin, we have and hence . Consequently, from Eq. (13), the expected scaling of is
For large but still smaller than the time to equilibrate the whole system, the dependence on must drop out and so
For short times where the spins in the window are random, so the mean square window overlap goes like (in dimensions) and consequently . Presumably we then have for . This actually gives but when corrections to scaling occur which cause to be replaced by a cutoff of order unity and so one obtains the desired result.
We take from Ref. Manssen and Hartmann (2014), evaluated according to Eq. (11), and also use the value of from Ref. Manssen and Hartmann (2014), . This exponent has also been computed in Ref. Belletti et al (2009) with a very similar value, . The result of scaling the data in Fig. 3 according to Eq. (14) is shown in the main part of Fig. 4. Clearly the scaling collapse works very well.
The power law decay of correlation in Eq. (6), and the resulting behavior of the window order parameter distribution in Eq. (14), are for a non-equilibrium situation where . However, we shall now see that a dramatic change occurs at sufficiently long times that global equilibrium occurs, i.e. when . In this region, we will find that the correlation function no longer decays to zero because there is spin glass order in equilibrium, and the weight of the window distribution at Marinari et al. (1998, 2000) becomes roughly independent of window size rather than increasing with window size in the manner shown in Eq. (14).
We demonstrate this change in behavior for the window overlaps explicitly as a function of time in Figs. 5 and 6. Since equilibrating size is completely infeasible we show data for smaller sizes which we can bring to global equilibrium. Figure 5 shows results for with window sizes and 8. A rapid decrease is seen for in the region – to a value which is independent of window size. As will be confirmed in Fig. 6, the data after the drop represents global equilibrium. The dashed line in Fig. 5 is the bulk value of the equilibrium overlap distribution at and we see that this value is very similar to that of equilibrium window overlaps, as was also found earlier Marinari et al. (1998, 2000).
To confirm that this change in behavior occurs when we plot results for for a fixed window size but different system sizes in Fig. 6. For short times the data is independent of indicating that , but at later times a more rapid decrease occurs at a time which increases with . The inset shows the data plotted against clearly demonstrating that the region with rapid decrease occurs when is about . This confirms that the decrease is associated with complete equilibration of the system.
For comparison we also did simulations of the ferromagnet, , at temperature . Since for the ferromagnet, this corresponds to , a similar fraction of as used in the spin glass simulations. The number of samples is detailed in Table 1. It should be pointed out that we are still using a single random number for multiple samples (due to multi-spin coding techniques) but with different initial configurations, as is common practice for spin-glass simulations. But in ferromagnetic equilibrium this causes the samples to become almost completely correlated. However, before equilibration, due to the different initial configurations, the dynamics of the different samples is different.
Data for the window overlap for window size and different lattice sizes are shown in Fig. 7. Even our largest systems can be equilibrated, as indicated by the data dropping to to a very small value () at the longest times. Note that the value of is not exactly zero even when the system has fully equilibrated because of rare thermal fluctuations. At short times the decay is roughly as expected from coarsening Bray (2002), according to which , the typical domain size, grows proportional to . However, in addition, a plateau appears at intermediate times. We shall see that this plateau occurs because in some runs, even when the correlation length has grown to the size of the system, a single domain with straight walls persists for a much longer time. Evidence for this is shown in the inset to Fig. 7 which plots the probability of finding two large clusters of oppositely oriented spins. This quantity has plateaus for the same range of time as the data for shown in the main part of the figure.
Figure 8 plots data for different system sizes and window sizes, and shows that the height of the plateau is proportional to which has a straightforward interpretation as the probability that a straight domain wall passes through the window.
We find that the time at the beginning of the plateau varies as , which is expected since it is the time for the coherence length to grow to the system size according to the coarsening picture in which . The time at the end of the plateau grows more rapidly and we find empirically it is roughly proportional to . We presume that this is the time scale needed for a random walk of the (straight) domain walls to cause the domains to meet and form one big domain. S. Redner (private communication) has argued that this exponent is exactly four, and our data is consistent with this value.
The rich dynamics of three-dimensional Ising ferromagnets after a quench have been studied in great detail, see e.g. Olejarz et al. (2011), at very low and zero temperature. By contrast, our results are for a much higher temperature, though still below . Based on the preliminary findings presented here, we feel it would be interesting to study this region in more detail in the future.
We have shown that the non-equilibrium window overlap distribution of a spin glass following a quench to below can be well characterized by the ratio of the dynamic coherence length to the window size . For a fixed the distribution tends to a well defined limit at long times such that but where is still much less than the system size . This distribution depends strongly on ; for example where .
However, if we can run the simulation for sufficiently long times that the system globally equilibrates, i.e. , then there is a change in behavior, which is abrupt when plotted on a logarithmic time scale, see Fig. 6, such that then only depends weakly on and is very similar to the value at zero overlap of Parisi’s global overlap distribution . Though a similar looking plateau was found for the not-quite fully equilibrated ferromagnet, characterized by the existence of domain walls, it was qualitatively different since the height of the plateau depends on the overall system size for the ferromagnet. By contrast, for the spin glass, the data in Fig. 6, while admittedly not fully in the plateau region, does not show any dependence on until the final equilibrium is reached (the end of the plateau).
The strong change in behavior for the spin glass when indicates that local spin correlations are sensitive to spin orientations, and presumably also to the values of the interactions, at large distances. According to the droplet theory, the local state of the system does not depend on the values of the interactions sufficiently far away. If the droplet theory is correct asymptotically, the length-scale beyond which this independence occurs must be much larger than the system sizes we have been able to equilibrate below (namely ).
In addition, we have speculated on a possible connection between the non-equilibrium dynamics discussed here and averages computed theoretically using the “metastate”. For a future better understanding of this possible connection via numerical simulations a more intense use of powerful yet rather cheap devices as GPUs, like in the present work Manssen and Hartmann (2014), or the application of new algorithms like population annealing to spin glasses Machta (2010); Wang et al. (2014) might be useful.
Acknowledgements.MM and AKH thank Martin Weigel for interesting discussions. We would like to thank Nick Read and David Huse for a most informative correspondence, and Sid Redner for suggesting the exact value of the exponent giving the length of the plateau in Fig. 8. The work of APY is supported in part by the National Science Foundation under Grant No. DMR-1207036, and by a Research Award from the Alexander von Humboldt Foundation.
- NS Newman and Stein (1996a, 1997a) call this the “non-standard” RSB picture, because they showed that a different, “standard”, RSB picture is not viable. As also emphasized recently by Read Read (2014), the “non-standard” picture is the only viable RSB picture, so we shall omit the term “non-standard” and just refer to this scenario as the “RSB picture”. In fact, Read also shows that the RSB calculations lead directly to the “non-standard” picture.
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