Replica symmetry breaking in and around six dimensions

TL;DR

This paper investigates replica symmetry breaking in six-dimensional Ising spin glasses, calculating key quantities and revealing new universal behaviors near the upper critical dimension, with implications for understanding spin glass phases.

Contribution

It provides the first calculations of the breakpoint x1 and Almeida-Thouless line in six dimensions, demonstrating the persistence and evolution of replica symmetry breaking around the critical dimension.

Findings

Replica symmetry breaking exists below d=6.

x1 has a universal nonzero value at criticality for d<6.

Near six dimensions, x1=3(6-d)+O[(6-d)^2].

Abstract

Two, replica symmetry breaking specific, quantities of the Ising spin glass --- the breakpoint x1 of the order parameter function and the Almeida-Thouless line --- are calculated in six dimensions (the upper critical dimension of the replicated field theory used), and also below and above it. The results comfirm that replica symmetry breaking does exist below d=6, and also the tendency of its escalation for decreasing dimension continues. As a new feature, x1 has a nonzero and universal value for d<6 at criticality. Near six dimensions we have x1=3(6-d)+O[(6-d)^2]. A method to expand a generic theory with replica equivalence around the replica symmetric one is also demonstrated.