Soft modes in vector spin glass models on sparse random graphs

We study numerically the Hessian of low-lying minima of vector spin glass models defined on random regular graphs. We consider the two-component (XY) and three-component (Heisenberg) spin glasses at zero temperature, subjected to the action of a randomly oriented external field. Varying the intensity of the external field, these models undergo a zero-temperature phase transition from a paramagnet at high field to a spin glass at low field. We study how the spectral properties of the Hessian depend on the magnetic field. In particular, we study the shape of the spectrum at low frequency and the localization properties of low-energy eigenvectors across the transition. We find that in both phases the edge of the spectral density behaves as ) 3/2: such a behavior rules out the presence of a diverging spin glass susceptibility XSG = (1/) 2). As to low-energy eigenvectors, we find that the softest eigenmodes are always localized in both phases of the two models. However, by studying in detail the geometry of low-energy eigenmodes across different energy scales close to the lower edge of the spectrum, we find a different behavior for the two models at the transition: in the XY case, low-energy modes are typically localized; at variance, in the Heisenberg case low-energy eigenmodes present a multimodal structure (sort of "delocalization"). These geometrically nontrivial excitations, which we call concentrated and delocalized low-energy modes, coexist with trivially localized excitations: we interpret their existence as a sign of critical behavior related to the onset of the spin glass phase.