Mode softening in time-crystalline transitions of open quantum systems

In this work, we generalize the concept of the roton-softening mechanism of the spatial crystalline transition to time crystals in open quantum systems. We study a dissipative Dicke model as a prototypical example, which exhibits both continuous time crystal and discrete time crystal phases. We find that, on approaching the time crystalline transition, the response function diverges at a finite frequency, which determines the period of the upcoming time crystal. This divergence can be understood as softening of the relaxation rate of the corresponding collective excitation, which can be clearly seen by the poles of the response function on the complex plane. Using this mode-softening analysis, we predict a time quasicrystal phase in our model, in which the self-organized period and the driving period are incommensurate.