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Discrete time crystals (DTCs) are new phases of matter characterized by the presence of an observable evolving with nT periodicity under a T -periodic Hamiltonian, where n > 1 is an integer insensitive to small parameter variations. In particular, DTCs with n = 2 have been extensively studied in periodically quenched and kicked spin systems in recent years. In this paper, we study the emergence of DTCs from the many-body quantum chaos perspective, using a rather simple model depicting a harmonically driven spin chain. We advocate to first employ a semiclassical approximation to arrive at a mean-field Hamiltonian and then identify the parameter regime at which DTCs exist, with standard tools borrowed from studies of classical chaos. Specifically, we seek symmetric-breaking solutions by examining the stable islands on the Poincaré surface of section of the mean-field Hamiltonian. We then turn to the actual many-body quantum system, evaluate the stroboscopic dynamics of the total magnetization in the full quantum limit, and verify the existence of DTCs. Our effective and straightforward approach indicates that in general DTCs are one natural aspect of many-body quantum chaos with mixed classical phase space structure.Our approach can also be applied to general time-periodic systems, which is thus promising for finding DTCs with n > 2 and opening possibilities for exploring DTCs properties beyond their time-translational breaking features.

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