The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

The ferromagnetic XY model on sparse random graphs in a randomly oriented field is analyzed via the belief propagation algorithm. At variance with the fully connected case and with the random field Ising model on the same topology, we find strong evidence of a tiny region with replica symmetry breaking (RSB) in the limit of very low temperatures. This RSB phase is robust against different choices of the external field direction, while it rapidly vanishes when increasing the graph mean degree, the temperature or the directional bias in the external field. The crucial ingredients to have such a RSB phase seem to be the continuous nature of vector spins, mostly preserved by the O(2)-invariant random field, and the strong spatial heterogeneity, due to graph sparsity. We also uncover that the ferromagnetic phase can be marginally stable despite the presence of the random field. Finally, we study the proper correlation functions approaching the critical points to identify the ones that become more critical.