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

## TL;DR

This study uses high temperature series expansions to analyze the Almeida-Thouless instability in short-range Ising spin-glasses across multiple dimensions, revealing scaling violations and supporting the existence of the AT line in high dimensions.

## Contribution

It provides the first detailed series expansion analysis of the AT line in finite-dimensional Ising spin-glasses, highlighting scaling violations and the dimensional dependence of the AT instability.

## Key findings

- AT line supported in dimensions d≥6
- Scaling violation in finite fields at high dimensions
- Larger critical exponents in d=5 compared to mean-field values

## Abstract

We use high temperature series expansions to study the $\pm J$ Ising spin-glass in a magnetic field in $d$-dimensional hypercubic lattices for $d=5, 6, 7$ and $8$, 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 $\Delta$ 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, there is a violation of scaling in a finite field, leading to an anomalous $h$-$T$ dependence of the Almeida-Thouless (AT) line in high dimensions, while 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 $\Delta$ are substantially larger than mean-field values, but we do not see clear evidence for the AT line in $d=5$.