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

Abstract

Liouvillian gap plays an important role in describing the asymptotic dynamics of quantum dissipative system. While the Liouvillian gap displays a power-law size scaling in boundary-dissipated systems with diffusive transport, its size-scaling relation is still not known in the presence of Anderson localization. In this letter, we study the Liouvillian gap of various one-dimensional quasiperiodic and disorder systems with boundary dissipation and unveil that Liouvillian gap fulfills an exponential scaling relation ∆g ∝ e −aL in the localized phase. By scrutinizing the extended Aubry-André-Harper model, we fit the value of a and find it coinciding pretty well with the analytical result of Lyapunov exponent κ which is the inverse of localization length ξ. We further apply the perturbation theory to derive ∆g ∝ e −L/ξ analytically. The exponential scaling relation was verified to hold true in other one-dimensional quasiperiodic and random disorder models. We also examine the relaxation time and show the inverse of Liouvillian gap giving a reasonable time scale for the system achieving the steady state.