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The ferromagnetic XY model on sparse random graphs in a randomly orientedfield is analyzed via the belief propagation algorithm. At variance with thefully connected case and with the random field Ising model on the sametopology, we find strong evidences of a tiny region with Replica SymmetryBreaking (RSB) in the limit of very low temperatures. This RSB phase is robustagainst different choices of the external field direction, while it rapidlyvanishes when increasing the graph mean degree, the temperature or thedirectional bias in the external field. The crucial ingredients to have such aRSB phase seem to be the continuous nature of vector spins, mostly preserved bythe O(2)-invariant random field, and the strong spatial heterogeneity, due tograph sparsity. We also uncover that the ferromagnetic phase can be marginallystable despite the presence of the random field. Finally, we study the propercorrelation functions approaching the critical points to identify the ones thatbecome more critical.

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