Replica Cluster Variational Method

TL;DR

This paper develops a formalism extending the Cluster Variational Method with replica symmetry breaking, applying it to spin glasses, and demonstrates improved stability and accuracy over Bethe approximation in 2D models.

Contribution

It introduces a hierarchical replica-symmetric and symmetry-breaking formalism within Kikuchi's CVM, providing a new variational approach for disordered systems.

Findings

Spurious spin-glass transition disappears in 2D with this method.

Paramagnetic phase remains stable down to zero temperature in 2D lattices.

Improved estimates of free energy compared to Bethe approximation.

Abstract

We present a general formalism to make the Replica-Symmetric and Replica-Symmetry-Breaking ansatz in the context of Kikuchi's Cluster Variational Method (CVM). Using replicas and the message-passing formulation of CVM we obtain a variational expression of the replicated free energy of a system with quenched disorder, both averaged and on a single sample, and make the hierarchical ansatz using functionals of functions of fields to represent the messages. We begin to study the method considering the plaquette approximation to the averaged free energy of the Edwards-Anderson model in the paramagnetic Replica-Symmetric phase. In two dimensions we find that the spurious spin-glass phase transition of the Bethe approximation disappears and the paramagnetic phase is stable down to zero temperature in all the three regular 2D lattices. The quantitative estimates of the free energy and of various other quantities improve those of the Bethe approximation. We provide the physical interpretation of the beliefs in the replica-symmetric phase as disorder distributions of the local Hamiltonians. The messages instead do not admit such an interpretation and indeed they cannot be represented as populations in the spin-glass phase at variance with the Bethe approximation.