Emergent Recurrent Extension Phase Transition in a Quasiperiodic Chain

We study $p$-wave superconducting quasiperiodic chains with staggered potential. The result shows a counter-intuitive phase transition phenomenon, i.e., recurrent extension phase transition (REPT). By analyzing the participation ration and scaling behavior, we prove the existence of REPT phenomenon, which, in concrete terms, means that the system will repeatedly return from the intermediate phase to the extended phase as the quasiperiodic or staggered strength grows. Furthermore, our finding is also quite different from the traditional understanding of intermediate phase (composed only of the pure extended phase and pure localized phase) in that, the new intermediate phase described here, stemming from the competition between staggered potential and $p$-wave pairing, actually falls into three types by bringing in the critical phase. To be specific, the new intermediate phases are composed of the critical + extended states, the critical + localized states, and the critical + extended + localized states, respectively.