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

Abstract

Mean field spin glass models have undergone substantial mathematical development, but finite dimensional short range spin glasses remain much less understood. This paper proves several rigorous zero temperature signatures of glassy behavior for the Edwards-Anderson model with Gaussian couplings, in finite boxes in arbitrary dimension. First, the ground state is sensitive to small perturbations of the disorder: after a perturbation of size $p$, the new ground state is nearly orthogonal to the original one in site overlap once $p$ is sufficiently larger than the inverse system size. Second, the droplets generated by such perturbations have large interfaces; in the macroscopic-droplet regime, their boundaries satisfy lower bounds consistent with a fractal dimension strictly greater than $d-1$. Third, there exist macroscopic spin excitations whose energy cost is negligible compared with the size of their interface, in sharp contrast with ferromagnetic behavior. Fourth, the expected size of the critical droplet associated with a typical bond grows at least as a power of the volume. Finally, a natural boundary condition sensitivity for nearest-neighbor spin products cannot decay faster than order the inverse distance to the boundary, contrasting with recent exponential decay results for the two-dimensional random field Ising model. Taken together, these results provide rigorous evidence -- and, in the senses made precise below, proof -- of zero temperature glassy behavior in a short range spin glass model.