Comparison of Gabay-Toulouse and de Almeida-Thouless instabilities for the spin glass XY model in a field on sparse random graphs

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.