Universality of dissipative discrete time crystal formation

We demonstrate that the Kibble-Zurek mechanism (KZM) holds for open systems transitioning from a disordered phase to a discrete time crystal (DTC). Specifically, we observe the characteristic power-law scaling with quench time of the number of spatial defects and the transition delay measured from the time at which the system crosses the critical point. We show analytically that this universal behavior can be traced back to how systems that can be mapped onto a dissipative linear parametric oscillator (DLPO) satisfy the adiabatic-impulse (AI) approximation, evinced by the divergence of the relaxation time of the DLPO near a critical point. We verify our predictions in both the classical and quantum regimes by considering two systems: the Sine-Gordon model, which is a paradigmatic system for emulating classical DTCs; and the open Dicke lattice model, an array of spin-boson systems subject to quantum fluctuations. We establish a universality class for DTC formation in systems that can be mapped onto a DLPO and show that the classical and quantum models considered here belong to this class.