Replica structure of one-dimensional disordered Ising models

– We analyse the eigenvalue structure of the replicated transfer matrix of onedimensional disordered Ising models. In the limit of n → 0 replicas, an infinite sequence of transfer matrices is found, each corresponding to a different irreducible representation (labelled by a positive integer ρ) of the permutation group. We show that the free energy can be calculated from the replica-symmetric subspace (ρ = 0). The other “replica symmetry broken” representations (ρ 6= 0) are physically meaningful, since their largest eigenvalues λ control the disorder-averaged moments 〈〈(〈SiSj〉−〈Si〉〈Sj〉)〉〉 ∝ (λ(ρ))|i−j| of the connected two-point correlations. In spite of extensive studies during the last twenty years, randomly disordered systems are not yet fully understood [1]. One of the main obstacles to the theoretical analysis of such systems is of course the lack of translational invariance of the Hamiltonian. The replica approach, which consists in replacing the original system and its randomly disordered Hamiltonian by the study of n (→ 0) identical and coupled systems with translationally invariant interactions, has been proven to be successful in providing a highly interesting mean-field theory of spin glasses [2], [3]. The latter relies on a fascinating, but mathematically obscure aspect of the replica mean-field theory, i.e. that the spin glass transition coincides with the so-called replica symmetry breaking (RSB), or, in other words, with the breaking of the permutation group symmetry of n→ 0 elements. In this context, the crucial question is to find the correct scheme of breaking, allowing an analytical continuation of the theory when the number n of replicas tends to zero. To what extent the physical picture originated in the resolution of long-range models [4] applies to finite-dimensional systems and the occurrence itself of replica symmetry breaking are still under question [5]. (∗) E-mail: [email protected] (∗∗) E-mail: [email protected] c © Les Editions de Physique 210 EUROPHYSICS LETTERS In this note, we present some preliminary remarks about this important issue by analysing one-dimensional spin models with quenched random couplings and/or fields. As in the case of one-dimensional ferromagnetism, no transition is expected at finite temperature [6]. However, since one-dimensional disordered systems may be solvable to a large extent with rigorous techniques, they constitute interesting examples on which the replica symmetry-breaking approach can be tested before being applied to more realistic systems. To start with, let us consider the Hamiltonian