Disappearance of the de Almeida-Thouless line in six dimensions

Abstract

We show that the Almeida-Thouless line in Ising spin glasses vanishes when their dimension $d\ensuremath{\rightarrow}{6}^{+}$ as ${h}_{\text{AT}}^{2}/{T}_{c}^{2}=C(d\ensuremath{-}6){}^{4}(1\ensuremath{-}T/{T}_{c}){}^{d/2\ensuremath{-}1}$, where $C$ is a constant of order unity. It is shown that replica symmetry breaking also stops as $d\ensuremath{\rightarrow}{6}^{+}$. Equivalent results that could be checked by simulations are given for the one-dimensional Ising spin glass with long-range interactions.