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It is proven rigorously that the ground state in the Edwards-Anderson spinglass model is unique in any dimension for almost all continuous randomexchange interactions under a condition that a single spin breaks the global${\mathbb Z}_2$ symmetry. This theorem implies that replica symmetry breakingdoes not occur at zero temperature. The site- and bond-overlap are concentratedat their maximal values. It is argued that behaviors of short range spin glassmodels are much different from those of mean field spin glass models near zerotemperature. Errata have been attached at the final page.

Uniqueness of ground state in the Edwards-Anderson spin glass model