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 $n \rightarrow 0$ 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.