# The realizable solutions of the TAP equations

## TL;DR

This paper demonstrates that iterative solutions of the TAP equations for the SK model only find states at the boundary of replica symmetry, which are marginally stable and exhibit self-organized criticality, revealing insights into the energy landscape.

## Contribution

It identifies the specific solutions of the TAP equations accessible via iteration as those at the replica symmetry boundary, linking stability, eigenvalue spectra, and criticality.

## Key findings

- Iterative solutions find states on the replica symmetry boundary.
- Solutions at this boundary are marginally stable with zero eigenvalues.
- The study links stability, free energy barriers, and self-organized criticality.

## Abstract

We show that the only solutions of the TAP equations for the Sherrington-Kirkpatrick model of Ising spin glasses which can be found by iteration are those whose free energy lies on the border between replica symmetric and broken replica symmetric states, when the number of spins $N$ is large. Convergence to this same borderline also happens in quenches from a high temperature initial state to a locally stable state where each spin is parallel to its local field; both are examples of self-organized criticality. At this borderline the band of eigenvalues of the Hessian associated with a solution extends to zero, so the states reached have marginal stability. We have also investigated the factors which determine the free energy difference between a stationary solution corresponding to a saddle point and its associated minimum, which is the barrier which has to be surmounted to escape from the vicinity of a TAP minimum or pure state.