A quantum phase of matter can be understood from the symmetry of the systems 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.
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