  Vector spin glasses are known to show two different kinds of phase
transitions in presence of an external field: the so-called de Almeida-Thouless
and Gabay-Toulouse lines. While the former has been studied to some extent on
several topologies (fully connected, random graphs, finite-dimensional
lattices, chains with long-range interactions), the latter has been studied
only in fully connected models, which however are known to show some unphysical
behaviors (e.g. the divergence of these critical lines in the zero-temperature
limit). Here we compute analytically both these critical lines for XY spin
glasses on random regular graphs. We discuss the different nature of these
phase transitions and the dependence of the critical behavior on the field
distribution. We also study the crossover between the two different critical
behaviors, by suitably tuning the field distribution.
