Localization Transition in Incommensurate non-Hermitian Systems

A class of one-dimensional lattice models with incommensurate complex potential $V(\theta )=2[{\lambda }_{r}cos(\theta )+i{\lambda }_{i}sin(\theta )]$ is found to exhibit localization transition at $\mathrm{\mid }{\lambda }_r\mathrm{\mid }+\mathrm{\mid }{\lambda }_i\mathrm{\mid }=1$. 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 $|\lambda_r| \ge |\lambda_i|$. 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}.