Low energy excitations in Mean Field Spin Glasses at zero temperature

The problem of low energy excitation at low temperatures of glassy system has arisen a great deal of interest in the last decades. The vibrational density of states of glassy systems (VDOS) is found to follow a seemingly universal quartic law approaching zero frequency. The corresponding eigenmodes are found to be localised. When phonons are present, these excitations are in excess with respect to the Debye prediction. Given the crucial importance of the soft modes of excitations for the low temperatures physics, it is most welcomed to achieve a theoretical understanding of this phenomenon.

In this talk, we analyse the problem in vector spin glass models: each spin is a vector with unitary modulus. We present results related to two Mean Field models: a long-range spin glass defined on a fully connected graph and a spin glass defined on a random regular graph. In both cases, the VDOS of the system at zero-temperature is computed, both analytically (when possible) and numerically, and the localisation properties of the eigenvectors are analysed. The system is studied varying the strength of an external random magnetic field.