On the formal equivalence of the TAP and thermodynamic methods in the SK model

TL;DR

This paper proves the formal equivalence between TAP and thermodynamic methods in the SK model using BRST supersymmetry, revealing that the TAP complexity aligns with the free energy calculation at one-step replica symmetry breaking.

Contribution

It demonstrates the equivalence of TAP and thermodynamic approaches in the SK model through BRST symmetry, correcting previous estimates of TAP complexity.

Findings

TAP and thermodynamic methods are formally equivalent.

The TAP complexity is smaller than previously estimated.

The equivalence is established at one-step replica symmetry breaking.

Abstract

We revisit two classic Thouless-Anderson-Palmer (TAP) studies of the Sherrington-Kirkpatrick model [Bray A J and Moore M A 1980 J. Phys. C 13, L469; De Dominicis C and Young A P, 1983 J. Phys. A 16, 2063]. By using the Becchi-Rouet-Stora-Tyutin (BRST) supersymmetry, we prove the general equivalence of TAP and replica partition functions, and show that the annealed calculation of the TAP complexity is formally identical to the quenched thermodynamic calculation of the free energy at one step level of replica symmetry breaking. The complexity we obtain by means of the BRST symmetry turns out to be considerably smaller than the previous non-symmetric value.