From scale-free to Anderson localization: a size-dependent transition

Scale-free localization in non-Hermitian systems is a distinctive type of localization where the localization length of certain eigenstates, known as scale-free localized (SFL) states, scales proportionally with the system size. Unlike skin states, where the localization length is independent of the system size, SFL states maintain a spatial profile that remains invariant as the system size changes. We consider a model involving a single non-Hermitian impurity in an otherwise Hermitian one-dimensional lattice. Introducing disorder into this system transforms SFL states into Anderson-localized states. In contrast to the Hatano-Nelson model, where disorder typically leads to the localization of skin states and a size-independent Anderson transition, the scale-free localization in our model causes a size-dependent Anderson transition.