Spectral evolution and real-space topological classification in a non-Hermitian Aubry–André–Harper model with quasiperiodic on-site potential

We present a systematic numerical study of the one-dimensional non-Hermitian Aubry–André–Harper model, which incorporates asymmetric nonreciprocal hopping and a quasiperiodic on-site potential. Using exact diagonalization, we investigate the interplay between the non-Hermitian skin effect driven by nonreciprocal hopping and Anderson localization induced by quasiperiodic disorder. We first analyze the evolution of the complex energy spectrum as the quasiperiodic potential strength V is tuned, revealing four characteristic stages: a fully closed point-gap topological loop in the complex plane, the onset of band splitting, the rupture of the topological loop into independent sub-loops, and the full collapse of the spectrum onto the real axis. We then construct detailed two-dimensional localization phase diagrams based on the average inverse participation ratio, demonstrating that nonreciprocal hopping enriches the phase structure of the standard Hermitian AAH model by introducing left-skin and right-skin topological phases alongside the extended metallic and Anderson insulating phases. Finally, we compute the real-space winding number under twisted boundary conditions and map out the complete topological phase diagram, clearly delineating regions with winding numbers W = +1, W = −1, and W = 0. Our results provide a comprehensive numerical characterization of the competition between quasiperiodic disorder and non-Hermitian nonreciprocity.