Disappearance of the de Almeida-Thouless line in six dimensions

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

Figure 1

The island of stability of the Gaussian fixed point for $d>6$. Distances are measured in units of $\sqrt{\epsilon }$. Only the region $w_2>0$ is displayed.