Symmetry-protected control of Liouvillian topological phases via Hamiltonian band topology

We establish a symmetry-protected correspondence between band topology of coherent Hamiltonians and Liouvillian spectral winding in Lindblad descriptions of open quantum systems. This allows the Hamiltonian topology to act as a knob for controlling Liouvillian topology and corresponding non-equilibrium dynamics, rather than being passively manipulated by system-environment exchanges. In particular, by exactly solving the Liouvillian spectrum in a class of one-dimensional dissipative lattices, we find that the Hamiltonian band topology constrains the Liouvillian spectral winding and determines the Liouvillian skin effect, provided the Hamiltonian and quantum jump operators respect the same chiral symmetry. We further demonstrate that lattice parity controls the associated bulk-boundary correspondence and the coherence properties of the steady state. Our results unveil a symmetry-enforced topological control of spectral and spatial organization in open quantum systems, providing a unified perspective on topology in Hamiltonian and dissipative dynamics.