  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 Poincare 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.
