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.