Topological phase transition in a generalized Aubry-Andre-Harper system with p -wave superconductivity

We study a variant of the generalized Aubry-Andre-Harper (GAAH) model with the effect of introducing the next
nearest-neighbor p-wave superconducting (SC) pairing with incommensurate and commensurate modulations
which are modulated by a cosine function. We demonstrate that the model consists of two superimposed two onedimensional
Majorana chains coupled with onsite potentials. We extend the GAAH model with p-wave SC to
topologically equivalent and nontrivial an “ancestor” two-dimensional p-wave SC model. It is found that in
incommensurate (commensurate) modulation, by varying the next nearest-neighbor p-wave pairing order
parameter, the system can switch between extended states and localized states (fully gapped phase and a gapless
phase). In the commensurate case, this model belongs to the AI mirror-symmetry class with time-reversal, particlehole,
chiral, and reflection symmetries. In the incommensurate case, we also numerically investigate the
localization properties of the GAAH model combined with the superconducting pairing. We show that localization
transition can be induced by varying just the phase, keeping all other model parameters constant. Also, in the
incommensurate case, we map the phase diagram between extended, localized, critical, and mobility edge phases.
We find a new phase, similar to the mobility edge phase, but with a less sudden rise between localized and
extended state than usual. We also find that value of Chern numbers are changing with and without on-site
potential when we tune the modulated p-wave SC pairing strength. We also calculate the Hofstadter butterflies for
the commensurate case.