Stiffness of the Edwards-Anderson Model in all Dimensions

A comprehensive description in all dimensions is provided for the scaling exponent $y$ of low-energy excitations in the Ising spin glass introduced by Edwards and Anderson. A combination of extensive numerical as well as theoretical results suggest that its lower critical dimension is exactly $d_l=5/2$. Such a result would be an essential feature of any complete theory of low-temperature spin glass order and imposes a constraint that may help to distinguish between theories.

Figure 1

Plot of the spin glass stiffness exponent $y_d$ as a function of dimension $d$. Shown are the data for hypercubic lattices from Refs. 23, 35, a weighted cubic fit to the $y_d$ data on $2\le d\le 6$, and the RSB prediction from Ref. 26 in Eq. (6). The fit predicts to 0.1% that the lower critical dimension defined through $y_{d_l}=0$ is $d_l=5/2$ (vertical line). It reproduces the exact result, $y_1=-1$ (checkered square), to 0.8% and suggests $y_0=-2$. Above the upper critical dimension, $d_u=6$, the data significantly deviates from Eq. (6) (solid line).

Figure 2

Plot of the glass-transition temperature $T_g(d)$ as a function of dimension $d$. Data points are drawn from various studies [28, 29, 30, 43, 44]. The solid line represents the fit in Eq. (5). That fit captures the known asymptotic behavior $T_g(d)\sim \sqrt{2d}$ for $d\to \infty$ and predicts a zero at $d\approx 2.491$.