  Density-matrix topology, defined through the geometric property of the
relevant modular Hamiltonian, can undergo transitions in the corresponding
open-system dynamics. While symmetry considerations are crucial to ensure such
a dynamic topological transition, we show that a hidden parity-time symmetry
can further facilitate it. Considering the Lindbladian dynamics of a fermionic
Gaussian state, we extract the time-evolved density-matrix topology from the
single-particle correlation, whose dynamics is governed by a non-Hermitian
damping matrix. We show that, for a parity-time symmetric damping matrix and a
chiral symmetric correlation matrix, a dynamic transition in the density-matrix
topology necessarily occurs in the parity-time unbroken regime where
eigenvalues of the damping matrix are real. We illustrate our results using a
concrete model, and map out the dynamic phase diagram.Remarkably, we find that
the dynamic transition can also happen periodically in the parity-time symmetry
broken regime.
