  A quantum phase of matter can be understood from the symmetry of the system's
Hamiltonian. The system symmetry along the time axis has been proposed to show
a new phase of matter referred as discrete-time crystals (DTCs). A DTC is a
quantum phase of matter in non-equilibrium systems, and it is also intimately
related to the symmetry of the initial state. DTCs that are stable in isolated
systems are not necessarily resilient to the influence from the external
reservoir. In this paper, we discuss the dynamics of the DTCs under the
influence of an environment. Specifically, we consider a non-trivial situation
in which the initial state is prepared to partly preserve the symmetry of the
Liouvillian. Our analysis shows that the entire system evolves towards a DTC
phase and is stabilised by the effect of dephasing. Our results provide a new
understanding of quantum phases emerging from the competition between the
coherent and incoherent dynamics in dissipative non-equilibrium quantum
systems.
