  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 Lindladian
parity-time (PT) symmetry can generically produce persistent periodic
oscillations, including dissipative continuous time crystals, in one-collective
spin models. By making an analogy to non-reciprocal phase transitions, we
demonstrate that a transition point from the dynamical phase is associated with
spontaneous PT symmetry breaking and typically corresponds to a
\textit{critical exceptional point}. Interestingly, the periodic orbits in the
PT-symmetric phase are found to be center-type, implying an
initial-state-dependent amplitude. These results are established by proving
that the Lindbladian PT symmetry at the microscopic level implies a non-linear
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.
