Liouvillian Skin Effect: Slowing Down of Relaxation Processes without Gap Closing

It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from spectral gap Liouvillian that governs time evolution density matrix. We investigate processes exhibits skin effect, which means eigenmodes are localized exponentially close boundary system, and find timescale for reach steady state depends not only on $\Delta$ but also localization length $\xi$ eigenmodes. In particular, show longest $\tau$ maximized over initial states local observables given by $\tau \sim \Delta^{-1}(1+L/\xi)$ with $L$ being size. This implies diverge $L \to \infty$ without closing.