Replica structure of one--dimensional Ising models

Abstract

We analyse the eigenvalue structure of the replicated transfer matrix of one-dimensional disordered Ising models. In the limit of $n\to 0$ 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 $\ll (⟨S_i{S}_j⟩-⟨S_i⟩⟨S_j⟩{)}^{\rho }\gg \propto ({\lambda }^{(\rho )}{)}^{|i-j|}$ of the connected two-points correlations.

4 pages, REVTeX, no figures

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