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arXiv:1406.1639 · Physical Review B

Cumulative overlap distribution function in realistic spin glasses

Alain H. Billoire, A. Maiorano, Enzo Marinari, Víctor Martín-Mayor, David Yllanes

Published Sep 8, 2014• 16 citations
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Abstract

We use a sample-dependent analysis, based on medians and quantiles, to analyze the behavior of the overlap probability distribution of the Sherrington-Kirkpatrick and $3D$ Edwards-Anderson models of Ising spin glasses. We find that this approach is an effective tool to distinguish between replica symmetry breaking--like and droplet-like behavior of the spin-glass phase. Our results are in agreement with a replica symmetry breaking--like behavior for the $3D$ Edwards-Anderson model.

DOI
10.1103/physrevb.90.094201

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We use a sample-dependent analysis, based on medians and quantiles, to analyze the behavior of the overlap probability distribution of the Sherrington-Kirkpatrick and 3D3D3D Edwards-Anderson models of Ising spin glasses. We find that this approach is an effective tool to distinguish between replica symmetry breaking--like and droplet-like behavior of the spin-glass phase. Our results are in agreement with a replica symmetry breaking--like behavior for the 3D3D3D Edwards-Anderson model.
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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 characterize numerically the properties of the phase transition of the three-dimensional Ising spin glass with Gaussian couplings and of the low-temperature phase. We compute critical exponents on large lattices. We study in detail the overlap probability distribution and the equilibrium overlap-overlap correlation functions. We find a clear agreement with off-equilibrium results from previous work. These results strongly support the existence of a continuous spontaneous replica symmetry breaking in three-dimensional spin glasses.
We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of L3{L}^{3}L3 spins. After averaging over large sets of EA system samples of $3\ensuremath{\le}L\ensuremath{\le}10$, we obtain accurate numbers for distributions p(q)p(q)p(q) of the overlap parameter qqq at very low-temperature TTT. We find $p(0)/T\ensuremath{\rightarrow}0.233(4)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links---between replica pairs---that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of $q\ensuremath{\sim}0$ excitations in the EA model ${d}_{s}\ensuremath{\rightarrow}2.59(3)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with ${d}_{s}\ensuremath{\rightarrow}3$ as $T\ensuremath{\rightarrow}0$ that is predicted by mean-field theory for the macroscopic limit.
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.