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Spontaneous breaking of continuous time translation symmetry into a discreteone is related to time crystal formation. While the phenomenon is not possiblein the ground state of a time-independent many-body system, it can occur in anexcited eigenstate. Here, we concentrate on bosons on a ring with attractivecontact interactions and analyze a quantum quench from the time crystal regimeto the non-interacting regime. We show that dynamical quantum phase transitionscan be observed where the return probability of the system to the initial statebefore the quench reveals a non-analytical behavior in time. The problem weconsider constitutes an example of the dynamical quantum phase transitions in asystem where both time and space continuous translation symmetries are broken.

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