Localization transition in incommensurate non-Hermitian systems

A class of one-dimensional lattice models with an incommensurate complex potential V(θ)=2[λᵣcos(θ)+iλᵢsin(θ)] is found to exhibit a localization transition at |λᵣ|+|λᵢ|=1. This transition from extended to localized states manifests itself in the behavior of the complex eigenspectum. In the extended phase, states with real eigenenergies have a finite measure, and this measure goes to zero in the localized phase. Furthermore, all extended states exhibit real spectra provided |λᵣ|>~|λᵢ|. Another interesting feature of the system is the fact that the imaginary part of the spectrum is sensitive to the boundary conditions only at the onset to localization.