Classical coupled parametric oscillators as an example of time crystals defined by order parameter dynamics

Discrete time crystals, which are phases of matter that break the discrete time translational symmetry of a periodically-driven system, have generated a wave of recent interest and activity. In this work, we propose a classical system of weakly nonlinear parametrically-driven coupled oscillators as a testbed to understand these phases. We show that this system belongs to a robust class of two-state time crystals defined by the criterion that over one time step, a total order parameter changes between two values with a difference proportional to the system size. Any system satisfying this requirement exhibits time-translation symmetry breaking that is robust to any perturbation as well as random noise that preserves the time-translation symmetry of the system. We then discuss applications of the general condition to existing time crystal platforms including quantum systems.