Twisted-boundary-condition characterization of discrete time crystals

In periodically driven systems, discrete time crystals break the discrete time-translation invariance of the driving field and oscillate with a period that is an integer multiple of the driving period [1]. Here, we impose a kind of twisted boundary condition [2], where one side of the boundaries is shifted along the time direction with respect to the other side, to capture the characteristics of the discrete time crystal phase. Specifically, we evaluate the spectral form factor of the kicked Ising model [3] in many-body localized phases under the twisted boundary condition. Coherent oscillation in the discrete time crystal is disturbed by the interaction across the boundaries oscillating at different phases. We found two types of signatures of the discrete time crystal, one in a short-time behavior in the thermodynamic limit, and the other in a long-time behavior of a small system.

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