  The interplay between dissipation, topology and sensitivity to boundary
conditions has recently attracted tremendous amounts of attention at the level
of effective non-Hermitian descriptions. Here we exactly solve a quantum
mechanical Lindblad master equation describing a dissipative topological
Su-Schrieffer-Heeger (SSH) chain of fermions for both open boundary condition
(OBC) and periodic boundary condition (PBC). We find that the extreme
sensitivity on the boundary conditions associated with the non-Hermitian skin
effect is directly reflected in the rapidities governing the time evolution of
the density matrix giving rise to a Liouvillian skin effect. This leads to
several intriguing phenomena including boundary sensitive damping behavior,
steady state currents in finite periodic systems, and diverging relaxation
times in the limit of large systems. We illuminate how the role of topology in
these systems differs in the effective non-Hermitian Hamiltonian limit and the
full master equation framework.
