The Complexity of Ising Spin Glasses

TL;DR

This paper analyzes the complexity of TAP states in Ising spin glasses, revealing the relationship between minima and saddle points, and showing that all states become marginally stable in the thermodynamic limit.

Contribution

It provides a detailed computation of the complexity of TAP states, including minima and saddle points, and confirms theoretical predictions about their stability.

Findings

Higher-index saddles have smaller complexities.

The two leading complexities are equal, consistent with Morse theorem.

All TAP states become marginally stable in the thermodynamic limit.

Abstract

We compute the complexity (logarithm of the number of TAP states) associated with minima and index-one saddle points of the TAP free energy. Higher-index saddles have smaller complexities. The two leading complexities are equal, consistent with the Morse theorem on the total number of turning points, and have the value given in [A. J. Bray and M. A. Moore, J. Phys. C 13, L469 (1980)]. In the thermodynamic limit, TAP states of all free energies become marginally stable.