Dynamical quantum phase transitions in systems with broken continuous time and space translation symmetries

Spontaneous breaking of continuous time translation symmetry into a discrete one is related to time crystal formation. While the phenomenon is not possible in the ground state of a time-independent many-body system, it can occur in an excited eigenstate. Here, we concentrate on bosons on a ring with attractive contact interactions and analyze a quantum quench from the time crystal regime to the noninteracting regime. We show that dynamical quantum phase transitions can be observed where the return probability of the system to the initial state before the quench reveals a nonanalytical behavior in time. The problem we consider constitutes an example of the dynamical quantum phase transitions in a system where both time and space continuous translation symmetries are broken.