Evidence of Non-Mean-Field-Like Low-Temperature Behavior in the Edwards-Anderson Spin-Glass Model

Abstract

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.

Numerical Construction of the Aizenman-Wehr Metastate

“…We also have two major theoretical frameworks that are applied to interpret experiments and simulations: the replica symmetry breaking (RSB) theory [15,16] and the droplet model [17][18][19]. Which (if any) of these two theories captures the nature of the spin-glass phase in d ¼ 3 is being debated [20].…”

“…Yet, the main debated points regard the equilibrium metastate. In fact, the only related issue addressed numerically by equilibrium simulations has been nonself-averageness [20,28,[31][32][33].…”

Extracting thermodynamic behavior of spin glasses from the overlap function

“…This statistic is a measure of the probability of peaks at small q < q 0 and is nearly independent of system size in 3D models. 17 In the 2D bimodal simulations, this lack of dependence on L appears to result from a combination of fewer peaks in P J (q) and a sharpening of the peaks as L increases. 18 The probability that a peak exceeds a threshold κ may then be relatively insensitive to L, though there is only one state.…”

“…For fixed q 0 and κ, this statistic can rise or decline with increasing L, due to the competition between the diminishing weight of peaks and the narrowing of the peaks. value of (q 0 ,κ,L) was found 17 to be roughly constant in L for small q 0 and κ for the three-dimensional Ising spin glass and to rise with L in the mean-field Sherrington-Kirkpatrick model. 4 The results of the simulations for two-dimensional Ising glass ground states are plotted in Fig.…”

Numerical results for the Edwards-Anderson spin-glass model at low temperature

“…Rather, w and δp become, within errors, independent of L in the zero-temperature limit. (The statistics of spikes in overlap distributions in different system samples, which have been studied by Yucesoy 6 et al, also point away from an RSB scenario, though this conclusion is criticized in Ref. 7.…”