Origin of spectral topology in non-Hermitian Aubry-André-Harper models

We investigate localization and topological properties of a non-Hermitian generalization of the Aubry-Andr\'e-Harper (AAH) model, which is the dimension reduction of the two-dimensional non-Hermitian quantum Hall model. Non-Hermiticity arises from the nonreciprocal hopping and the complex global phase. In the absence of nonreciprocal hopping, by applying Avila's global theory we analytically compute the localization length of states and determine phase diagrams. The complex global phase is in favor of the critical phase, whose transition to the localized phase falls outside the universality class of the classic AAH model. By mapping to the Hall model, we further demonstrate that the loop structure in spectrum originates from the spectral topology in the reduced dimension. When the nonreciprocal hopping is added, the localization is analytically studied by employing a similarity transformation, and phase diagrams are determined exactly. The system has boundary-dependent spectra and states because of the nontrivial spectral topology in the remaining physical dimension. The critical phase no longer exists, and the nonreciprocal hopping favors the extended phase when under the periodic boundary condition. Loops always exist in the complex spectrum but have distinct topological origins in different phases.