Spin glass phase at zero temperature in the Edwards-Anderson model

Abstract

While the analysis of mean-field spin glass models has seen tremendous progress in the last twenty years, lattice spin glasses have remained largely intractable. This article presents the solutions to a number of questions about the Edwards-Anderson model of short-range spin glasses (in all dimensions) that were raised in the physics literature many years ago. First, it is shown that the ground state is sensitive to small perturbations of the disorder, in the sense that a small amount of noise gives rise to a new ground state that is nearly orthogonal to the old one with respect to the site overlap inner product. Second, it is shown that one can overturn a macroscopic fraction of the spins in the ground state with an energy cost that is negligible compared to the size of the boundary of the overturned region -a feature that is believed to be typical of spin glasses but clearly absent in ferromagnets. The third result is that the boundary of the overturned region in dimension d has fractal dimension strictly greater than d − 1, confirming a prediction from physics. The fourth result is that the correlations between bonds in the ground state can decay at most like the inverse of the distance. This contrasts with the random field Ising model, where it has been shown recently that the correlation decays exponentially in distance in dimension two. The fifth result is that the expected size of the critical droplet of a bond grows at least like a power of the volume. Taken together, these results comprise the first mathematical proof of glassy behavior in a short-range spin glass model.

An Introduction to the Theory of Spin Glasses

“…Evidence in favor of the RSB picture is, however, not decisive, and relevant beyond-MF mechanisms might arise in low dimensions. Alternative or intermediate scenarios could also be valid [146], such as the trivial-non-trivial picture [161,162], according to which the domain surfaces are not space-filling, as in the Droplet picture, but there are large-scale excitations whose energy does not increase with size [163,164], as prescribed by the RSB scenario.…”

Superconcentration for minimal surfaces in first passage percolation and disordered Ising ferromagnets

“…We shall illustrate this in Corollary 6.1 where we will rely on a mild extension of a very useful identity from [26]. See also the recent work of Chatterjee [9] which analyzed the groundstate of an Ising model with non-ferromagnetic disordered coupling constants.…”

Some Rigorous Results on the Lévy Spin Glass Model

“…The study of spin glasses has drawn attention from both physicists and mathematicians for decades, in particular, a lot of efforts have been made to understand the Sherrington-Kirkpatrick mean-field model [45] as well as the Edwards-Anderson short-range model [24]. Whereas the investigation on the former has achieved tremendous success (see, e.g., [37,40,46,47]), the latter was scarcely understood due to its rigid geometry until recent years, see [15] and the references therein. In the hope of bridging the gap between the mean-field model and realistic models of short-range interactions, diluted variants of the Sherrington-Kirkpatrick model have been introduced, such as the spin glass model on a Bethe lattice, the Viana-Bray model, and the diluted p-spin models (see [32,36,49]) that possess a bounded average number of spin interactions at any site as opposed to the diverging connectivity in the fully-connected models.…”