Analytical mobility edge in nonreciprocal quasiperiodic lattices with next-nearest-neighbor hopping

This paper investigates the localization transitions and spectral topology of a one-dimensional non-Hermitian generalization of the Aubry-André model. Specifically, the authors examine how nonreciprocal hopping—where particles move more easily in one direction than the other—affects the mobility edge (ME) that separates extended states from localized states in a quasiperiodic potential.

To solve for the mobility edge, the authors extend the Fermi-surface point-matching method to non-Hermitian systems. By applying a Bloch ansatz to the nonreciprocal Hamiltonian, they demonstrate that nonreciprocity parameters (β and δ) act as imaginary shifts to the momentum, effectively renormalizing the hopping amplitudes into exponentially larger values. This allows them to map the localization boundary as a parabola in the energy-potential plane. They validate this analytical framework using exact diagonalization across various configurations of nearest-neighbor and next-nearest-neighbor nonreciprocity.

The study reveals that the mobility edge forms a single parabola in the energy-potential plane. Nearest-neighbor nonreciprocity rigidly shifts the localization boundary toward stronger potentials, while next-nearest-neighbor nonreciprocity reduces the curvature of the boundary, thereby broadening the energy window in which extended and localized states coexist. Furthermore, the authors show that spectral winding numbers—a topological diagnostic for the non-Hermitian skin effect—can be used to bracket the mixed phase. As the potential strength increases, the winding number at the lower band edge drops when the first localized states appear, and the winding number at the upper band edge drops when the last extended states localize.

This work provides a compact, analytical framework that connects energy-dependent localization, spectral topology, and nonreciprocity. Because the derived mobility edge is expressed in a closed form, it offers a predictive tool for understanding how open-system effects like gain, loss, or nonreciprocal transport influence the fundamental physics of Anderson localization. These results are directly testable in experimental platforms such as photonic lattices, cold atomic gases, and topolectrical circuits.