Topological phases induced by the Aubry-André-Harper potential in the longer-range Kitaev superconducting chain

We investigate the one-dimensional longer-range Kitaev chain with an on-site complex Aubry-Andre-Harper (AAH) potential. For the incommensurate (commensurate) case, by calculating the real-space (Bloch) winding number and energy spectrum we discover the AAH potential-induced phases and present the related phase diagrams. We also find that the imaginary phase shift in AAH potentials can suppress these topological phases for both cases. For the incommensurate case, the inverse participate ratios of the eigenstates with the smallest |E|s indicate the existence of the incommensurate AAH potential-induced phases, and the result of the mean inverse participate ratio shows that these incommensurate AAH potential-induced phases are not always strongly localized.