Non-Hermitian skin effect as an impurity problem

A striking feature of non-Hermitian tight-binding Hamiltonians is the high sensitivity of both spectrum and eigenstates to boundary conditions. Indeed, if the spectrum under periodic boundary conditions is point gapped by opening the lattice the non-Hermitian skin effect will necessarily occur. Finding the exact skin eigenstates may be demanding in general, and many methods in the literature are based on recurrence equations for the eigenstates' components. Here we devise a general procedure based on the Green's function method to calculate the eigenstates of non-Hermitian tight-binding Hamiltonians under open boundary conditions. We apply it to the Hatano-Nelson and non-Hermitian Su-Schrieffer-Heeger models and finally we contrast the edge states localization with that of bulk states.