Controlling Liouvillian topological phases via Hamiltonian band topology under chiral symmetry

We establish a direct connection between Hamiltonian band topology and the Liouvillian spectral winding of open quantum systems with quadratic dissipations. This allows the band topology to act as a knob for controlling Liouvillian topology and corresponding non-equilibrium dynamics. In particular, we show analytically that, for chiral Hamiltonians under sublattice-unidirectional chiral dissipation, the Liouvillian spectral winding depends on the Hamiltonian solely through its band winding number, thereby enabling direct band-topological control of the Liouvillian skin effect. We further identify two distinguished coherent and incoherent Liouvillian skin effects therein: the former hosts a band-topological dark state of the Hamiltonian, whereas the latter yields a mixed steady state driven by dissipative nonreciprocity. Our results establish a systematic framework for topological control of spectral and spatial organization in open quantum systems and provide a unified perspective on topology in Hamiltonian and dissipative dynamics.