The classical mutual information in mean-field spin glass models

TL;DR

This paper studies the classical Renyi entropy and mutual information in the mean-field spin glass Sherrington-Kirkpatrick model, revealing a glassy phase, a paramagnetic phase, and unique finite-size effects at the transition.

Contribution

It provides analytical and numerical analysis of mutual information and entropy in the SK model, including the effects of replica symmetry breaking and the geometry of the system.

Findings

Identifies a glassy phase with replica symmetry breaking.

Shows mutual information follows a volume law across phases.

Finds no crossing of mutual information at the critical point for different system sizes.

Abstract

We investigate the classical Renyi entropy S_n and the associated mutual information I_n in the Sherrington-Kirkpatrick (S-K) model, which is the paradigm model of mean-field spin glasses. Using classical Monte Carlo simulations and analytical tools we investigate the S-K model on the n-sheets booklet. This is obtained by gluing together n independent copies of the model, and it is the main ingredient to construct the Renyi entanglement-related quantities. We find a glassy phase at low temperature, whereas at high temperature the model exhibits paramagnetic behavior, consistent with the regular S-K model. The temperature of the paramagnetic-glassy transition depends non-trivially on the geometry of the booklet. At high-temperatures we provide the exact solution of the model by exploiting the replica symmetry. This is the permutation symmetry among the fictitious replicas that are used to perform disorder averages (via the replica trick). In the glassy phase the replica symmetry has to be broken. Using a generalization of the Parisi solution, we provide analytical results for S_n and I_n, and for standard thermodynamic quantities. Both S_n and I_n exhibit a volume law in the whole phase diagram. We characterize the behavior of the corresponding densities S_n/N, I_n/N, in the thermodynamic limit. Interestingly, at the critical point the mutual information does not exhibit any crossing for different system sizes, in contrast with local spin models.