Aubry-Andre-Harper Model with on-site hopping and p-wave pairing*

We study an extended Aubry-André-Harper model with simultaneous modulation of hopping on-site potential and p-wave superconducting pairing. For the case of commensurate modulation of β=1/2 it is shown that the model hosts four different types of topological states: Adiabatic cycles can be defined which pump particles two types of Majorana fermions or Cooper pairs. In the incommensurate case we calculate the phase diagram of the model in several regions. We characterize the phases by calculating the mean inverse participation ratio and perform multifractal analysis. In addition we characterize whether the phases found are topologically trivial or not. We find an interesting critical extended phase when incommensurate hopping modulation is present. The rise between the inverse participation ratio in regions separating localized and extended states is gradual rather than sharp. When in addition the on-site potential modulation is incommensurate we find several sharp rises and falls in the inverse participation ratio. In these two cases all different phases exhibit topological edge states.