Localization transition in incommensurate non-Hermitian systems

Abstract

A class of one-dimensional lattice models with an incommensurate complex potential $V(\ensuremath{\theta})=2[{\ensuremath{\lambda}}_{r}\mathrm{cos}(\ensuremath{\theta})+i{\ensuremath{\lambda}}_{i}\mathrm{sin}(\ensuremath{\theta})]$ is found to exhibit a localization transition at $|{\ensuremath{\lambda}}_{r}|+|{\ensuremath{\lambda}}_{i}|=1.$ 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 $|{\ensuremath{\lambda}}_{r}|>~|{\ensuremath{\lambda}}_{i}|.$ 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.