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 $\xi(t) \propto t^{1/z(T)}$ 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 $d_u$, for both free and periodic boundary conditions. For $d>d_u$, 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 $\vec{k}=\vec{0}$ fluctuations, while standard FSS applies to $\vec{k}\neq\vec{0}$ 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 $T-T_L$, where $T_L$ is the finite-size pseudocritical temperature, as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at $T_L$, where the susceptibility $\chi$ diverges like $L^{d/2}$, and the bulk $T_c$, where it diverges like $L^2$. We support these claims with data from large-scale simulations of the five-dimensional Ising model.
