On the TAP free energy in the mixed $p$-spin models

TL;DR

This paper establishes that in mixed p-spin models, the free energy can be expressed as a supremum of the TAP free energy over certain magnetizations, linking it to the Parisi measure and pure states.

Contribution

It proves the representation of free energy as a supremum of TAP free energy in mixed p-spin models and connects it to the Parisi measure and pure states.

Findings

Free energy equals the supremum of TAP free energy over specific magnetizations.

In generic models, free energy matches TAP free energy at pure states.

The TAP free energy representation holds for mixed p-spin models with Ising spins.

Abstract

In [Physical Magazine, 35(3):593-601, 1977], Thouless, Anderson, and Palmer derived a representation for the free energy of the Sherrington-Kirkpatrick model, called the TAP free energy, written as the difference of the energy and entropy on the extended configuration space of local magnetizations with an Onsager correction term. In the setting of mixed $p$-spin models with Ising spins, we prove that the free energy can indeed be written as the supremum of the TAP free energy over the space of local magnetizations whose Edwards-Anderson order parameter (self-overlap) is to the right of the support of the Parisi measure. Furthermore, for generic mixed $p$-spin models, we prove that the free energy is equal to the TAP free energy evaluated on the local magnetization of any pure state.