Anderson localization in non-hermitian systems: skin effect and scale-free localization

Abstract

Localization properties of eigenstates change according to the lattice model. A Hatano-Nelson model with asymmetric hopping constants exhibit non-Hermitian skin effect(NHSE) where eigenstates localize near the edge of the lattice, the localization length is finite and does not depend on the system size. With introducing disorder to the Hatano-Nelson model, skin states start to become Anderson localized states. At a critical disorder strength all skin states become Anderson localized states. Since localization length of skin states does not depend on the system size, this model exhibits a size-independent Anderson localization. When the Hatano-Nelson model is modified as a 1D Hermitian lattice with an impurity, new model exhibits scale-free localized (SFL) states where unlike skin states, localization length scales with the system size. SFL states have a spatial profile that does not change as the system size varies. With the introduction of disorder into the system with SFL states, the interplay between SFL and Anderson localized states is observed. At the critical disorder strength, all states become Anderson localized states. Since localization length of SFL states scales with system size, a size-dependent Anderson transition occurs.