Vector spin glass models have been scarcely studied, mostly on fully connected topologies, which unfortunately provide some unphysical predictions. We focus instead on diluted topologies, which yield more reliable results and whose predictions should be closer to what actually happens in the finite-dimensional case. By exploiting the belief-propagation algorithm, we first of all show that the XY model (the simplest vector model) can be both reliably and efficiently approximated via its discrete version (the Q-state clock model) in numerical simulations. Then, we show that the diluted spin glass XY model is by far more unstable toward replica symmetry breaking (RSB) with respect to discrete models (e.g. Ising spins), due to the combined effect of the graph sparsity and the spin continuous nature. Several interesting consequences arise, among which the appearance of a RSB phase in the random field XY model, at variance forbidden in the random field Ising model, and the marginal stability of the low-temperature phase. Moreover, the spin glass XY model in a random field can be analytically solved down to the zero-temperature limit, allowing the exact computation of the ground state and the characterization of the energy landscape. Its critical properties are believed to be strongly related to that of finite-dimensional structural glasses, so representing a reliable tool to obtain a meaningful, self-consistent description of the low-temperature behavior of glasses. First, important evidences of such analogy have been already observed in the density of the soft modes characterizing the system at the critical point, having the same exponent of the frequency density of vibrations in glasses, at variance with respect to any other mean-field approach exploited so far.
The joint presence of both continuous variables and sparse graphs has not been exploited at all so far in the field of disordered systems, so we believe that this research could provide remarkable progress in this field. This is confirmed by the interest exhibited by several international collaborators, during the exhibition of the preliminary results.
In particular, it is striking how this model represents the first analytically solvable, nontrivial theoretical model that correctly reproduces the density of soft modes in the low-temperature regime of glass formers. Similar results have been obtained so far only via approximated studies, while mean-field approaches turn out to provide some predictions that do not match when moving from the infinite-dimension limit to finite dimensions. If confirmed, our approach would provide a rather different and new perspective from which study the behavior of glasses.
Moreover, the techniques we are developing for vector models on diluted graphs could find interesting applications in related fields, such as computer science, inference and learning.