Evidence that the AT transition disappears below six dimensions

TL;DR

The study investigates the existence of the de Almeida-Thouless (AT) transition in spin glasses across different dimensions, finding that it disappears below six dimensions, challenging the applicability of Parisi's theory in three dimensions.

Contribution

The paper provides numerical evidence that the AT transition vanishes below six dimensions, indicating limitations of Parisi's replica symmetry breaking scheme in lower-dimensional spin glasses.

Findings

AT line vanishes below six dimensions

Numerical simulations show $h_{AT}^2$ scales with $(2/3 - \sigma)$

Parisi scheme may not apply to 3D spin glasses

Abstract

One of the key predictions of Parisi's broken replica symmetry theory of spin glasses is the existence of a phase transition in an applied field to a state with broken replica symmetry. This transition takes place at the de Almeida-Thouless (AT) line in the $h-T$ plane. We have studied this line in the power-law diluted Heisenberg spin glass in which the probability that two spins separated by a distance $r$ interact with each other falls as $1/r^{2\sigma}$. In the presence of a random vector-field of variance $h_r^2$ the phase transition is in the universality class of the Ising spin glass in a field. Tuning $\sigma$ is equivalent to changing the dimension $d$ of the short-range system, with the relation being $d =2/(2\sigma -1)$ for $\sigma < 2/3$. We have found by numerical simulations that $h_{\text{AT}}^2 \sim (2/3 -\sigma)$ implying that the AT line does not exist below $6$ dimensions and that the Parisi scheme is not appropriate for spin glasses in three dimensions.