Exponential size scaling of the Liouvillian gap in boundary-dissipated systems with Anderson localization

We carry out a systematical study of the size scaling of the Liouvillian gap in boundary-dissipated one-dimensional quasiperiodic and disorder systems. By treating the boundary-dissipation operators as a perturbation, we derive an analytical expression of the Liouvillian gap, which indicates clearly the Liouvillian gap being proportional to the minimum of boundary densities of eigenstates of the underlying Hamiltonian, and thus give a theoretical explanation why the Liouvillian gap has different size scaling relation in the extended and localized phase. While the Liouvillian gap displays a power-law size scaling ${\mathrm{\ensuremath{\Delta}}}_{g}\ensuremath{\propto}{L}^{\ensuremath{-}3}$ in the extended phase, our analytical result unveils that the Liouvillian gap fulfills an exponential scaling relation ${\mathrm{\ensuremath{\Delta}}}_{g}\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\kappa}L}$ in the localized phase, where $\ensuremath{\kappa}$ takes the largest Lyapunov exponent of localized eigenstates of the underlying Hamiltonian. By scrutinizing the extended Aubry-Andr\'e-Harper model, we numerically confirm that the Liouvillian gap fulfills the exponential scaling relation and the fitting exponent $\ensuremath{\kappa}$ coincides pretty well with the analytical result of the Lyapunov exponent. The exponential scaling relation is further verified numerically in other one-dimensional quasiperiodic and random disorder models. We also study the relaxation dynamics and show the inverse of the Liouvillian gap giving a reasonable timescale of asymptotic convergence to the steady state.