Localization Transition in Incommensurate non-Hermitian Systems

Abstract

A class of one-dimensional lattice models with incommensurate complex potential $V(θ)=2[λ_r cos(θ)+i λ_i sin(θ)]$ is found to exhibit localization transition at $|λ_r|+|λ_i|=1$. This transition from extended to localized states manifests in the behavior of the complex eigenspectum. In the extended phase, states with real eigenenergies have finite measure and this measure goes to zero in the localized phase. Furthermore, all extended states exhibit real spectrum provided $|λ_r| \ge |λ_i|$. Another novel feature of the system is the fact that the imaginary part of the spectrum is sensitive to the boundary conditions {\it only at the onset to localization}.