The Complexity of Vector Spin Glasses

TL;DR

This paper analyzes the complexity of vector spin glasses, revealing spectral properties of the Hessian matrix and demonstrating supersymmetry breaking similar to Ising spin glasses.

Contribution

It provides a detailed spectral analysis of the Hessian in vector spin glasses and shows supersymmetry breaking occurs in these models, extending previous Ising results.

Findings

Eigenvalue spectrum includes a continuous positive band and an isolated eigenvalue.

The continuous band does not reach zero at finite temperature.

The isolated eigenvalue vanishes in the thermodynamic limit, indicating supersymmetry breaking.

Abstract

We study the annealed complexity of the m-vector spin glasses in the Sherrington-Kirkpatrick limit. The eigenvalue spectrum of the Hessian matrix of the Thouless-Anderson-Palmer (TAP) free energy is found to consist of a continuous band of positive eigenvalues in addition to an isolated eigenvalue and (m-1) null eigenvalues due to rotational invariance. Rather surprisingly, the band does not extend to zero at any finite temperature. The isolated eigenvalue becomes zero in the thermodynamic limit, as in the Ising case (m=1), indicating that the same supersymmetry breaking recently found in Ising spin glasses occurs in vector spin glasses.