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arXiv:1408.0438 · Physical Review B

Evidence against a mean-field description of short-range spin glasses revealed through thermal boundary conditions

Wenlong Wang, Jonathan L. Machta, Helmut G. Katzgraber

Published Nov 12, 2014• 32 citations
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Abstract

A theoretical description of the low-temperature phase of short-range spin glasses has remained elusive for decades. In particular, it is unclear if theories that assert a single pair of pure states, or theories that are based on infinitely many pure states---such as replica symmetry breaking---best describe realistic short-range systems. To resolve this controversy, the three-dimensional Edwards-Anderson Ising spin glass in thermal boundary conditions is studied numerically using population annealing Monte Carlo. In thermal boundary conditions all eight combinations of periodic vs antiperiodic boundary conditions in the three spatial directions appear in the ensemble with their respective Boltzmann weights, thus minimizing finite-size corrections due to domain walls. From the relative weighting of the eight boundary conditions for each disorder instance a sample stiffness is defined, and its typical value is shown to grow with system size according to a stiffness exponent. An extrapolation to the large-system-size limit is in agreement with a description that supports the droplet picture and other theories that assert a single pair of pure states. The results are, however, incompatible with the mean-field replica symmetry breaking picture, thus highlighting the need to go beyond mean-field descriptions to accurately describe short-range spin-glass systems.

DOI
10.1103/physrevb.90.184412

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A theoretical description of the low-temperature phase of short-range spin glasses has remained elusive for decades. In particular, it is unclear if theories that assert a single pair of pure states, or theories that are based on infinitely many pure states---such as replica symmetry breaking---best describe realistic short-range systems. To resolve this controversy, the three-dimensional Edwards-Anderson Ising spin glass in thermal boundary conditions is studied numerically using population annealing Monte Carlo. In thermal boundary conditions all eight combinations of periodic vs antiperiodic boundary conditions in the three spatial directions appear in the ensemble with their respective Boltzmann weights, thus minimizing finite-size corrections due to domain walls. From the relative weighting of the eight boundary conditions for each disorder instance a sample stiffness is defined, and its typical value is shown to grow with system size according to a stiffness exponent. An extrapolation to the large-system-size limit is in agreement with a description that supports the droplet picture and other theories that assert a single pair of pure states. The results are, however, incompatible with the mean-field replica symmetry breaking picture, thus highlighting the need to go beyond mean-field descriptions to accurately describe short-range spin-glass systems.
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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=8d=8d=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)\\xi(t)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)\\xi(t) \\propto t^{1/z(T)}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 ddd above the upper critical dimension dud_udu​, for both free and periodic boundary conditions. For d>dud>d_ud>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\\vec{k}=\\vec{0}veck=vec0 fluctuations, while standard FSS applies to veckneqvec0\\vec{k}\\neq\\vec{0}veckneqvec0 fluctuations. Hence the exponent eta\\etaeta describing the power-law decay of correlations at criticality is unambiguously eta=0\\eta=0eta=0. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using T−TLT-T_LT−TL​, where TLT_LTL​ is the finite-size pseudocritical temperature, as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at TLT_LTL​, where the susceptibility chi\\chichi diverges like Ld/2L^{d/2}Ld/2, and the bulk TcT_cTc​, where it diverges like L2L^2L2. We support these claims with data from large-scale simulations of the five-dimensional Ising model.
The relative importance of the contributions of droplet excitations and domain walls on the ordering of short-range Edwards-Anderson spin glasses in three and four dimensions is studied. We compare the spin overlap distribution functions of periodic and free boundary conditions using population annealing Monte Carlo. For system sizes up to about 1000 spins, spin glasses show nontrivial spin overlap distributions. Periodic boundary conditions may trap diffusive domain walls which can contribute to small spin overlaps, and the other contribution is the existence of low-energy droplet excitations within the system. We use free boundary conditions to minimize domain-wall effects, and show that low-energy droplet excitations are the major contribution to small overlaps in numerical simulations. Free boundary conditions has stronger finite-size effects, and is likely to have the same thermodynamic limit with periodic boundary conditions.
We study the fragility of spin glasses to small temperature perturbations numerically using population annealing Monte Carlo. We apply thermal boundary conditions to a three-dimensional Edwards-Anderson Ising spin glass. In thermal boundary conditions all eight combinations of periodic versus antiperiodic boundary conditions in the three spatial directions are present, each appearing in the ensemble with its respective statistical weight determined by its free energy. We show that temperature chaos is revealed in the statistics of crossings in the free energy for different boundary conditions. By studying the energy difference between boundary conditions at free-energy crossings, we determine the domain-wall fractal dimension. Similarly, by studying the number of crossings, we determine the chaos exponent. Our results also show that computational hardness in spin glasses and the presence of chaos are closely related.
Using Monte Carlo simulations, we study in detail the overlap distribution for individual samples for several spin-glass models including the infinite-range Sherrington-Kirkpatrick model, short-range Edwards-Anderson models in three and four space dimensions, and one-dimensional long-range models with diluted power-law interactions. We study three long-range models with different powers as follows: The first is approximately equivalent to a short-range model in three dimensions, the second to a short-range model in four dimensions, and the third to a short-range model in the mean-field regime. We study an observable proposed earlier by some of us which aims to distinguish the ``replica symmetry breaking'' picture of the spin-glass phase from the ``droplet picture,'' finding that larger system sizes would be needed to unambiguously determine which of these pictures describes the low-temperature state of spin glasses best, except for the Sherrington-Kirkpatrick model, which is unambiguously described by replica symmetry breaking. Finally, we also study the median integrated overlap probability distribution and a typical overlap distribution, finding that these observables are not particularly helpful in distinguishing the replica symmetry breaking and the droplet pictures.
We present a collection of simulations of the Edwards–Anderson lattice spin glass at T=0 to elucidate the nature of low-energy excitations over a range of dimensions that reach from physically realizable systems to the mean-field limit. Using heuristic methods, we sample the ground states of instances to determine their energies while eliciting excitations through manipulating boundary conditions. We exploit the universality of the phase diagram of bond-diluted lattices to make such a study in higher dimensions computationally feasible. As a result, we obtain a variety of accurate exponents for domain wall stiffness and finite-size corrections, which allow us to examine their dimensional behavior and their connection with predictions from mean-field theory. We also provide an experimentally testable prediction for the thermal-to-percolative crossover exponent in dilute lattice Ising spin glasses.