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arXiv:cond-mat/0503155 · Physical Review B

Factorization properties in the three-dimensional Edwards-Anderson model

Pierluigi Contucci, Cristian Giardinà

Published Jul 21, 2005• 7 citations
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Abstract

We study the three-dimensional Gaussian Edwards-Anderson model and find numerical evidence of a simple factorization law of the link-overlaps distributions at large volumes. We also perform the same analysis for the standard overlap for which instead the lack of factorization persists, increasing the size of the system. Our results open new perspectives in the study of the two different overlaps emphasizing the importance of the concept of factorization-triviality to distiniguish their role.

DOI
10.1103/physrevb.72.014456

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We study the three-dimensional Gaussian Edwards-Anderson model and find numerical evidence of a simple factorization law of the link-overlaps distributions at large volumes. We also perform the same analysis for the standard overlap for which instead the lack of factorization persists, increasing the size of the system. Our results open new perspectives in the study of the two different overlaps emphasizing the importance of the concept of factorization-triviality to distiniguish their role.
We study the relative fluctuations of the link overlap and the square standard overlap in the three-dimensional Gaussian Edwards-Anderson model with zero external field. We first analyze the correlation coefficient and find that the two quantities are uncorrelated above the critical temperature. Below the critical temperature we find that the link overlap has vanishing fluctuations for fixed values of the square standard overlap and large volumes. Our data show that the conditional variance scales to zero in the thermodynamic limit. This implies that, if one of the two random variables tends to a trivial one (i.e., deltalike distributed), then the other does also, and as a consequence, the "trivial-nontrivial" picture should be dismissed. Our results show that the two overlaps are completely equivalent in the description of the low temperature phase of the Edwards-Anderson model.
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 investigate the distributions of the link overlap, P(Q)P(Q)P(Q), in three-dimensional Ising spin glasses. We use clustering methodology to identify a set of pairs of states from different Gibbs states and calculate its contribution to P(Q)P(Q)P(Q). We find that the distribution over this set does not become trivial as the system size increases.
We numerically extract large-scale excitations above the ground state in the 3-dimensional Edwards-Anderson spin glass with Gaussian couplings. We find that associated energies are O(1), in agreement with the mean field picture. Of further interest are the position-space properties of these excitations. First, our study of their topological properties show that the majority of the large-scale excitations are sponge-like. Second, when probing their geometrical properties, we find that the excitations coarsen when the system size is increased. We conclude that either finite size effects are very large even when the spin overlap q is close to zero, or the mean field picture of homogeneous excitations has to be modified.