Delocalization transition in low energy excitation modes of vector spin glasses

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