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arXiv:2005.05805 · Physical Review Research

Evidence of many thermodynamic states of the three-dimensional Ising spin glass

Wenlong Wang, Mats Wallin, Jack Lidmar

Published Nov 16, 2020• 6 citations
arXivHTMLPDF

Abstract

We present a large-scale simulation of the three-dimensional Ising spin glass with Gaussian disorder to low temperatures and large sizes using optimized population annealing Monte Carlo.Our primary focus is investigating the number of pure states regarding a controversial statistic, characterizing the fraction of centrally peaked disorder instances, of the overlap function order parameter.We observe that this statistic is subtly and sensitively influenced by the slight fluctuations of the integrated central weight of the disorder-averaged overlap function, making the asymptotic growth behavior very difficult to identify.Modified statistics effectively reducing this correlation are studied, and essentially monotonic growth trends are obtained.The effect of temperature is also studied, finding a larger growth rate at a higher temperature.Our state-of-the-art simulation and variance reduction data analysis suggest that the many pure states picture is most likely and coherent.

DOI
10.1103/physrevresearch.2.043241

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We present a large-scale simulation of the three-dimensional Ising spin glass with Gaussian disorder to low temperatures and large sizes using optimized population annealing Monte Carlo.Our primary focus is investigating the number of pure states regarding a controversial statistic, characterizing the fraction of centrally peaked disorder instances, of the overlap function order parameter.We observe that this statistic is subtly and sensitively influenced by the slight fluctuations of the integrated central weight of the disorder-averaged overlap function, making the asymptotic growth behavior very difficult to identify.Modified statistics effectively reducing this correlation are studied, and essentially monotonic growth trends are obtained.The effect of temperature is also studied, finding a larger growth rate at a higher temperature.Our state-of-the-art simulation and variance reduction data analysis suggest that the many pure states picture is most likely and coherent.
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.
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.
We have studied the three-dimensional Ising spin glass with a $\ifmmode\pm\else\textpm\fi{}J$ distribution by Monte Carlo simulations. Using larger sizes and much better statistics than in earlier work, a finite-size scaling analysis shows quite strong evidence for a finite transition temperature Tc{T}_{c}Tc​ with ordering below Tc{T}_{c}Tc​. Our estimate of the transition temperature is rather lower than in earlier work, and the value of the correlation length exponent $\ensuremath{\nu}$ is somewhat higher. Because there may be (unknown) corrections to finite-size scaling, we do not completely rule out the possibility that Tc=0{T}_{c}=0Tc​=0 or that Tc{T}_{c}Tc​ is finite but with no order below Tc{T}_{c}Tc​. However, from our data, these possibilities seem less likely.
Domain growth after a quench into the ordered phase is studied for the simple-cubic nearest-neighbor Gaussian Ising spin-glass model. A suitably defined time-averaged local overlap between two replicas is used to locate the domains and domain walls. The logarithmic domain growth rate dR(t)/d(logt), appears to increase with time t where R(t) is a measure of the linear size of the compact domains that are present. The effective fractal dimensionality of the domain walls is estimated. The bearing of the results on the many states versus two states controversy is briefly discussed.
We present a large-scale simulation of the three-dimensional and mean-field spin glasses down to a very low but finite temperature. We extrapolate pertinent observables, e.g., the disorder-averaged central weight to zero temperature, finding that many thermodynamic states at a finite temperature and two ground states at zero temperature are fully compatible. While the disorder-averaged central weight monotonically decreases with decreasing temperature, this is far from true for individual samples. This motivates us to link this behaviour with the well-known temperature chaos. At an observing temperature, a sample may or may not have pure state coexistence depending on whether it is undergoing temperature chaos, which is a random process. Therefore, temperature chaos is likely responsible for the emergence of many pure states, providing a natural and intuitive explanation for the coexistence of expensive domain-wall excitations and many pure states at the disorder-averaged level.
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.