Replica structure of one--dimensional Ising models

TL;DR

This paper investigates the eigenvalue structure of the replicated transfer matrix in one-dimensional disordered Ising models, revealing how different symmetry representations influence free energy calculations and correlation functions.

Contribution

It introduces an infinite sequence of transfer matrices corresponding to permutation group representations, clarifying the role of replica symmetry and symmetry breaking in disordered systems.

Findings

Free energy derived from the replica symmetric subspace.

Eigenvalues associated with symmetry-broken representations govern correlation moments.

Identification of physically meaningful eigenvalues controlling disorder-averaged correlations.

Abstract

We analyse the eigenvalue structure of the replicated transfer matrix of one-dimensional disordered Ising models. In the limit of $n \rightarrow 0$ replicas, an infinite sequence of transfer matrices is found, each corresponding to a different irreducible representation (labelled by a positive integer $\rho$) of the permutation group. We show that the free energy can be calculated from the replica symmetric subspace ($\rho =0$). The other ``replica symmetry broken'' representations ($\rho \ne 0$) are physically meaningful since their largest eigenvalues $\lambda^{(\rho )}$ control the disorder--averaged moments $\ll( \langle S_i S_j \rangle - \langle S_i \rangle \langle S_j \rangle )^\rho \gg \propto (\lambda^{(\rho )}) ^{|i-j|}$ of the connected two-points correlations.