On Supersymmetry Breaking in the Computation of the Complexity

TL;DR

This paper investigates how supersymmetry breaking affects the calculation of the number of solutions to TAP equations in spin glass models, extending previous arguments and analyzing eigenvalues related to supersymmetry.

Contribution

It extends Kurchan's argument to solutions at any free energy and proves the zero value of a specific eigenvalue due to supersymmetry breaking in the BM theory.

Findings

The prefactor of the Bray and Moore saddle point vanishes for all solutions.

The isolated eigenvalue is exactly zero because of supersymmetry breaking.

The eigenvector at the lower band edge exhibits specific behavior related to supersymmetry.

Abstract

We study the consequences of supersymmetry breaking in the computation of the number of solutions of the Thouless-Anderson-Palmer (TAP) equations. We show that Kurchan argument that proves the vanishing of the prefactor of the Bray and Moore saddle point for the total number of solutions can be extended to solutions at any given free energy. We also provide a new simple argument for the vanishing of the prefactor and use it to prove that the isolated eigenvalue recently considered by Aspelmeier, Bray and Moore is exactly zero in the BM theory because of supersymmetry breaking. The behavior of the eigenvector of the isolated eigenvalue at the lower band edge is also considered.