Localization transitions and winding numbers for non-Hermitian Aubry-André-Harper models with off-diagonal modulations

Abstract

We study localization and topological properties, as well as the fate of the critical phase, of non-Hermitian generalizations of the Aubry-Andr\'e-Harper model with both on-site and off-diagonal incommensurate modulations. Non-Hermiticity arises from nonreciprocal hopping and a complex phase in the potential. In the absence of nonreciprocal hopping, we compute analytically the localization length of single-particle states by applying Avila's global theory. The system has the same phase diagram as Hermitian cases, except that the complex phase renormalizes the strength of the potential. In the presence of nonreciprocal hopping, the phase diagram is analytically determined by a similarity transformation. Due to the presence of the skin effect, induced by nonreciprocity, the skin phase turns into extended and critical phases when the boundary condition changes from open to periodic, while states are localized asymmetrically in the boundary-independent localized phase. The nonreciprocal hopping is in favor of the critical phase under a periodic boundary condition. The spectra are complex, and loops always exist. A winding number is not a proper indicator of the presence of loops, but the topological transition is in agreement with the localization transition.