Boundary sensitive Lindbladians and relaxation dynamics

It is well known that non-Hermitian systems can be extremely sensitive to boundary conditions owing to non-Hermitian skin effect (NHSE). Analogously, we investigate two boundary-sensitive $U(1)$ symmetric Lindbladians: one carries current in the steady state, and the other does not. The numerical results indicate significant change of the Liouvillian spectrum, eigenmodes and relaxation time for both Lindbladians when the boundary conditions are altered. This phenomenon is found to be triggered by the Liouvillian skin effect (LSE), specifically the localization of eigenmodes, which stems from the NHSE of the non-Hermitian effective Hamiltonian. In addition, these two Lindbladians manifest different LSE, ultimately resulting in distinct relaxation behaviors.