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 non-interacting 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 non-analytical 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.