Analyticity of the energy in an Ising spin glass with correlated disorder

TL;DR

This paper generalizes the understanding of the analyticity of average energy in Ising spin glasses from uncorrelated to correlated disorder, showing no singularity along certain phase diagram lines despite phase transitions.

Contribution

It extends the known results on energy analyticity to correlated disorder cases and derives an exact expression for a related physical quantity in three dimensions.

Findings

Average energy has no singularity along specific phase diagram lines.

Derived an exact closed-form expression for a related physical quantity in 3D.

Proved identities and inequalities for specific heat and correlation functions.

Abstract

The average energy of the Ising spin glass is known to have no singularity along a special line in the phase diagram although there exists a critical point on the line. This result on the model with uncorrelated disorder is generalized to the case with correlated disorder. For a class of correlations in disorder that suppress frustration, we show that the average energy in a subspace of the phase diagram is expressed as the expectation value of a local gauge variable of the $Z_2$ gauge Higgs model, from which we prove that the average energy has no singularity although the subspace is likely to have a phase transition on it. This is a generalization of the result for uncorrelated disorder. Though it is difficult to obtain an explicit expression of the energy in contrast to the case of uncorrelated disorder, an exact closed-form expression of a physical quantity related to the energy is derived in three dimensions using a duality relation. Identities and inequalities are proved for the specific heat and correlation functions.