Incompleteness of the Thouless, Anderson, and Palmer mean-field description of the spin-glass phase

TL;DR

This paper critically examines the TAP mean-field equations for spin glasses, revealing their instability at low temperatures due to free energy degeneracy, and proposes an extended, stable framework using spin replicas.

Contribution

It introduces a replica-based extension of the TAP approach that ensures stability and homogeneity in describing the spin-glass phase.

Findings

TAP states are unstable at low temperatures due to degeneracy.

Infinitesimal interactions couple TAP solutions, affecting stability.

The extended framework provides a consistent, stable description of the spin-glass phase.

Abstract

We analyze the low-temperature behavior of mean-field equations of Thouless, Anderson, and Palmer (TAP). We demonstrate that degeneracy in free energy makes the low-temperature TAP states unstable. Different solutions of the TAP equations, independent in the TAP approach, become coupled if an infinitesimal interaction between them is introduced. By means of real spin replicas we derive a self-averaging free energy free of unstable states with local magnetizations and homogeneous overlap susceptibilities between different spin replicas as order parameters. We thereby extend the TAP approach to a consistent description of the spin-glass phase for all configurations of spin couplings with (marginally) stable and thermodynamically homogeneous free energy.