We study the energy minima of the fully-connected
mm-components
vector spin glass model at zero temperature in an external magnetic
field for m\ge 3m≥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}λm−1
in the paramagnetic phase and as \sqrt{\lambda}λ
at criticality and 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 2048N≤2048
and compute the spectral properties, that show very strong corrections
to the asymptotic scaling approaching the critical point.
Zero-temperature quantum annealing bottlenecks in the spin-glass phase
