Continuous time crystals as a PT symmetric state and the emergence of critical exceptional points

Continuous time-translation symmetry is often spontaneously broken in open quantum systems, and the condition for their emergence has been actively investigated. However, there are only a few cases in which its condition for appearance has been fully elucidated. In this Letter, we show that a Lindbladian parity-time ($\mathcal{PT}$) symmetry can generically produce persistent periodic oscillations in a wide class of systems. This includes one-collective spin models, which have been studied thoroughly in the context of dissipative continuous time crystals, and spatially extended bipartite bosonic systems with conserved particle number. Interestingly, the periodic orbits in the PT-symmetric phase are found to be center-type, implying an initial state dependence. These results are established by proving that the Lindbladian $\mathcal{PT}$ symmetry at the microscopic level implies a nonlinear PT symmetry, and by performing a linear stability analysis near the transition point. This research will further our understanding of novel non-equilibrium phases of matter and phase transitions with spontaneous anti-unitary symmetry breaking.