Delocalization transition in low energy excitation modes of vector spin glasses

TL;DR

This paper investigates the spectral properties and localization-delocalization transition of low-energy excitations in a vector spin glass model at zero temperature, revealing a transition from localized to delocalized eigenstates at the paramagnetic to spin glass phase boundary.

Contribution

It provides a detailed analysis of the eigenvalue spectrum and eigenvector localization in the vector spin glass, introducing the delocalization transition at the spectral edge as a key feature.

Findings

Spectrum is gapless in both phases with distinct pseudo-gap behaviors.

Eigenstates near the spectral edge are quasi-localized and become delocalized at the transition.

Numerical results show strong finite-size effects near the critical point.

Abstract

We study the energy minima of the fully-connected $m$-components vector spin glass model at zero temperature in an external magnetic field for $m\ge 3$. The model has a zero temperature transition from a paramagnetic phase at high field to a spin glass phase at low field. We study the eigenvalues and eigenvectors of the Hessian in the minima of the Hamiltonian. The spectrum is gapless both in the paramagnetic and in the spin glass phase, with a pseudo-gap behaving as $\lambda^{m-1}$ in the paramagnetic phase and as $\sqrt{\lambda}$ in the spin glass phase. Despite the long-range nature of the model, the eigenstates close to the edge of the spectrum display quasi-localization properties. We show that the paramagnetic to spin glass transition corresponds to delocalization of the edge eigenvectors. We solve the model by the cavity method in the thermodynamic limit. We also perform numerical minimization of the Hamiltonian for $N\le 2048$ and compute the spectral properties, that show very strong corrections to the asymptotic scaling approaching the critical point.