Is the droplet theory for the Ising spin glass inconsistent with replica field theory?

TL;DR

This paper uses symmetry arguments to derive exact identities in the replica symmetric field theory of the Ising spin glass, revealing stability properties and the need for resummation in different dimensions.

Contribution

It introduces exact identities between vertex functions applicable across various dimensions, clarifying the stability and phase distinctions in the Ising spin glass theory.

Findings

Replica symmetric theory is unstable for d>8.

Resummation is needed for 6<d<8 and d<6.

Ward-like identities help distinguish phases.

Abstract

Symmetry arguments are used to derive a set of exact identities between irreducible vertex functions for the replica symmetric field theory of the Ising spin glass in zero magnetic field. Their range of applicability spans from mean field to short ranged systems in physical dimensions. The replica symmetric theory is unstable for d>8, just like in mean field theory. For 6<d<8 and d<6 the resummation of an infinite number of terms is necessary to settle the problem. When d<8, these Ward-like identities must be used to distinguish an Almeida-Thouless line from the replica symmetric droplet phase.