Despite decades of effort, our understanding of low-temperature phase of spin glass models with short-range interactions remains incomplete. Replica symmetry breaking (RSB) theory, based on the solution of the Sherrington-Kirkpatrick mean-field model, predicts many pure states; meanwhile, competing theories of short-range systems, such as the droplet picture, predict a single pair of pure states related by time-reversal symmetry, analogously to the ferromagnet. Since RSB certainly holds for the mean-field (infinite-range) model, it is interesting to study short-range models in high dimensions to observe whether RSB also holds here; however, computer simulations of short-range models in high dimensions are difficult because the number of spins to equilibrate grows so rapidly with the linear size of the system.A relatively recent idea which has been fruitful is to instead study one-dimensional models with long-range (power-law) interactions, which are argued to have the same critical behavior as corresponding short-range models in high dimensions, but for which simulations for a range of sizes (crucial for finite-size scaling analysis) are feasible. For these one-dimensional long-range (1DLR) models, we fill in a previously unexplored region of parameter space where the interactions become sufficiently long-range that they must be rescaled with the system size to maintain the thermodynamic limit. We find strong evidence that detailed behavior of the 1DLR models everywhere in this "nonextensive regime" is identical to that of the Sherrington-Kirkpatrick model, lending support to a recent conjecture.In an attempt to distinguish the RSB and droplet pictures, we study recently-proposed observables based on the statistics of individual disorder samples, rather than simply averaging over the disorder as is most frequently done in previous studies. We compare Monte Carlo results for 1DLR models which are proxies for short-range models in 3, 4, and 10 dimensions with previously-obtained data for the 3D and 4D short-range models and the SK model. For one statistic, which is expected to sharply distinguish between the two pictures in the thermodynamic limit, we find that larger system sizes than those currently feasible to simulate are needed to obtain an unambiguous result. We also find that two other recently-proposed statistics, the median of the cumulative overlap distribution and the "typical" overlap distribution, are not particularly helpful in distinguishing between the RSB and droplet pictures.If there are many pure states in the spin-glass phase, we need to carry out some sort of statistical average over them to obtain the thermodynamics. One such prescription for doing this is called the "metastate." Motivated by similarities between the average over pure states specified by the metastate theory and that presumably generated by the nonequilibrium dynamics, we study a 1DLR model which is a proxy for a short-range model in
d=8 dimensions and measure the evolution of dynamical correlations. We find that the spatial decay of the correlations at distances less than the dynamical correlation length
xi(t) agrees quantitatively with the predictions of the metastate theory, evaluated according to the RSB picture. We also compute the dynamic exponent defined by
xi(t)proptot1/z(T) and find that it is compatible with the mean-field value of the critical dynamical exponent for short-range spin glasses.Finally, we present a unified view of finite-size scaling (FSS) in dimensions
d above the upper critical dimension
du, for both free and periodic boundary conditions. For
d>du, a dangerous irrelevant variable is responsible for both the violation of hyperscaling and the violation of "standard" FSS. We find that the modified hyperscaling proposed to allow for this applies only to
veck=vec0 fluctuations, while standard FSS applies to
veckneqvec0 fluctuations. Hence the exponent
eta describing the power-law decay of correlations at criticality is unambiguously
eta=0. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using
T−TL, where
TL is the finite-size pseudocritical temperature, as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at
TL, where the susceptibility
chi diverges like
Ld/2, and the bulk
Tc, where it diverges like
L2. We support these claims with data from large-scale simulations of the five-dimensional Ising model.
The three-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied via large-scale Monte Carlo simulations at low temperatures, deep within the spin-glass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states.
Spin glasses have been the subject of intense study and considerable controversy for decades, and the low-temperature phase of short-range spin glasses is still poorly understood. Our main goal is to improve our understanding in this area and find an answer to the following question: Are there only a single pair or a countable infinity of pure states in the low temperature phase of the EA spin glass? To that aim we first start by introducing spin glasses and provide a brief history of their research, then proceed to describe our method of simulation, the parallel tempering Monte Carlo algorithm. Next, we present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. In order to understand how the parallel tempering algorithm works, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. We examine the correlation between static and dynamic observables for _ 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature, and our results show that autocorrelation times are directly correlated with the roughness of the free-energy landscape. In the following chapters, the three- and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied again via large-scale Monte Carlo simulations at low temperatures, deep within the spinglass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. We arrive to the following conclusion: The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. Finally we present results for several new observables, along with a few preliminary studies and suggestions for future research.
We have argued in recent papers that the Monte Carlo results for the equilibrium properties of the Edwards-Anderson spin glass in three dimensions, which had been interpreted earlier as providing evidence for replica symmetry breaking, can be explained quite simply within the droplet model once finite-size effects and proximity to the critical point are taken into account. In this paper we show that similar considerations are sufficient to explain the Monte Carlo data in four dimensions. In particular, we study the Parisi overlap and the link overlap for the four-dimensional Ising spin glass in the Migdal-Kadanoff approximation. Similar to what is seen in three dimensions, we find that temperatures well below those studied in the Monte Carlo simulations have to be reached before the droplet model predictions become apparent. We also show that the double-peak structure of the link overlap distribution function is related to the difference between domain-wall excitations that cross the entire system and droplet excitations that are confined to a smaller region.
The question of the number of thermodynamic states present in the low-temperature phase of the three-dimensional Edwards-Anderson Ising spin glass is addressed by studying spin- and link-overlap distributions using population annealing Monte Carlo simulations. We consider overlaps between systems with the same boundary condition, which are the usual quantities measured, and also overlaps between systems with different boundary conditions, both for the full systems and also within a smaller window within the system. Our results appear to be fully compatible with a single pair of pure states such as in the droplet-scaling picture. However, our results for whether or not domain walls induced by changing boundary conditions are space filling or not are also compatible with scenarios having many thermodynamic states, such as the chaotic-pairs picture and the replica-symmetry-breaking picture. The differing results for spin overlaps in the same and different boundary conditions suggest that finite-size effects are very large for the system sizes currently accessible in low-temperature simulations.