  Discrete time crystals are related to non-equilibrium dynamics of
periodically driven quantum many-body systems where the discrete time
translation symmetry of the Hamiltonian is spontaneously broken into another
discrete symmetry. Recently, the concept of phase transitions has been extended
to non-equilibrium dynamics of time-independent systems induced by a quantum
quench, i.e. a sudden change of some parameter of the Hamiltonian. There, the
return probability of a system to the ground state reveals singularities in
time which are dubbed dynamical quantum phase transitions. We show that the
quantum quench in a discrete time crystal leads to dynamical quantum phase
transitions where the return probability of a periodically driven system to a
Floquet eigenstate before the quench reveals singularities in time. It
indicates that dynamical quantum phase transitions are not restricted to
time-independent systems and can be also observed in systems that are
periodically driven. We discuss how the phenomenon can be observed in
ultra-cold atomic gases.
