On Ward-Takahashi identities for the Parisi spin glass

TL;DR

This paper derives Ward-Takahashi identities for the Parisi spin glass model using small permutations, revealing Goldstone modes and relations between multi-point functions in the replica symmetry breaking limit.

Contribution

It introduces a novel method using small permutations to derive Ward-Takahashi identities in the Parisi limit of spin glasses, highlighting new symmetry-related insights.

Findings

Emergence of a band of Goldstone modes.

Relations between 3-point and 2-point functions.

Identification of a jump in the inverse propagator when overlaps cross.

Abstract

The introduction of ``small permutations'' allows us to derive Ward-Takahashi identities for the spin-glass, in the Parisi limit of an infinite number of steps of replica symmetry breaking. The first identities express the emergence of a band of Goldstone modes. The next identities relate components of (the Replica Fourier Transformed) 3-point function to overlap derivatives of the 2-point function (inverse propagator). A jump in this last function is exhibited, when its two overlaps are crossing each other, in the special simpler case where one of the cross-overlaps is maximal.