Emergence of spatial patterns and synchronization in superconducting time crystals

We identify a time crystal phase characterized by a frequency half of the driving frequency in disordered superconductors by employing the time-dependent Bogoliubov--de Gennes formalism at zero temperature with a periodically driven coupling constant. After a period of exponential increase of spatial inhomogeneities and exponential suppression of the order parameter amplitude, the time crystal develops islands of different sizes. Each of these islands is a time crystal with the same frequency albeit with a phase shift $\ensuremath{\pi}$ with respect to the homogeneous time crystal. After its emergence, the island gradually becomes smaller, though the phase shift persists, until it is abruptly synchronized at a time that it depends on its initial size. We find a critical disorder strength, still deep in the metallic phase, at which the time crystal phase terminates. For even stronger disorder, the order parameter oscillates with the driving frequency in regions where localization effects are not important.