de Almeida–Thouless instability in short-range Ising spin glasses

We use high-temperature series expansions to study the $\pm J$ Ising spin glass in a magnetic field in $d$-dimensional hypercubic lattices for $d=5–8$ and in the infinite-range Sherrington-Kirkpatrick (SK) model. The expansions are obtained in the variable $w={tanh}^{2}J/T$ for arbitrary values of $u={tanh}^{2}h/T$ complete to order $w^{10}$. We find that the scaling dimension $\mathrm{\Delta }$ associated with the ordering-field $h^2$ equals 2 in the SK model and for $d\ge 6$. However, in agreement with the work of Fisher and Sompolinsky [Phys. Rev. Lett. 54, 1063 (1985)], there is a violation of scaling in a finite field, leading to an anomalous $h\text{−}T$ dependence of the de Almeida–Thouless (AT) [J. Phys. A 11, 983 (1978)] line in high dimensions, whereas scaling is restored as $d\to 6$. Within the convergence of our series analysis, we present evidence supporting an AT line in $d\ge 6$. In $d=5$, the exponents $\gamma$ and $\mathrm{\Delta }$ are substantially larger than mean-field values, but we do not see clear evidence for the AT line in $d=5$.

Figure 1

Histogram of the exponent $\mathrm{\Delta }$ in zero field obtained from various approximants in (a) $d=8$, (b) $d=7$, (c) $d=6$, and (d) $d=5$ to the series for $K_1\left(D_1\right)$ and $K_2\left(D_2\right)$ defined in Eqs. (6) and (7).

Figure 2

The inverse of the spin-glass susceptibility in the SK model for (a) $u\equiv {tanh}^2(h/T)=0.1$ and (b) $u=0.2$. The AT line is where ${\chi }_{\mathrm{SG}}^{-1}=0$. The exact values obtained numerically from the equations of AT are shown by the solid red (thick) lines. Also shown are the (5, 5) (green, solid lines) and (4, 4) (blue, dashed lines) Padé approximants of the high-temperature series using the transformed variable $w={tanh}^2(1/T)$. It is seen that the divergence is located reasonably well at $u=0.1$ but less so at $u=0.2$.

Figure 3

The solid (blue) line is the expression for the AT line in the SK model, whereas the dashed (green) line is its asymptotic small $h$ limit. The points joined by lines are the results of the Padé analysis of the ten-term series, including just the Padé approximants that are bunched together.

Figure 4

Estimates of the AT line in $d=6,d=7$, and $d=8$ obtained from Padé analyses of the series. The formulas for the lines are given in the legend and are discussed in the text.

Figure 5

Approximants showing divergence of the spin-glass susceptibility in $d=5$ and the range of their exponent $\gamma$.