Stability of the replica-symmetric saddle-point in general mean-field spin-glass models

Details Stability of the replica-symmetric saddle-point in general mean-field spin-glass models Stability of the replica-symmetric saddle-point in general mean-field spin-glass models

2010-08-10

arxiv.org/abs/1008.1733v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

Scientific paper

Within the replica approach to mean-field spin-glasses the transition from ergodic high-temperature behaviour to the glassy low-temperature phase is marked by the instability of the replica-symmetric saddle-point. For general spin-glass models with non-Gaussian field distributions the corresponding Hessian is a $2^n\times 2^n$ matrix with the number $n$ of replicas tending to zero eventually. We block-diagonalize this Hessian matrix using representation theory of the permutation group and identify the blocks related to the spin-glass susceptibility. Performing the limit $n\to 0$ within these blocks we derive expressions for the de~Almeida-Thouless line of general spin-glass models. Specifying these expressions to the cases of the Sherrington-Kirkpatrick, Viana-Bray, and the L\'evy spin glass respectively we obtain results in agreement with previous findings using the cavity approach.

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