Topological superconductors and exact mobility edges in non-Hermitian quasicrystals

Abstract

We study a class of non-Hermitian topological superconductors described by one-dimensional Aubry-Andr\'e Harper and mosaic quasiperiodic models with $p$-wave superconducting pairing, where the non-Hermiticity is introduced by on-site complex quasiperiodic potentials. We generalize two topological invariants, one is based on the transfer matrix method and the other is the generalized Majorana polarization, to characterize the topological superconducting phases and verify the existence of Majorana zero modes in non-Hermitian quasiperiodic superconductors. By combing the Lyapunov exponent, the fractional dimension of wave functions, and topological invariants, we investigate the localization phenomena, topological superconductivity, and topological phase transitions. In the non-Hermitian Aubry-Andr\'e Harper model with $p$-wave pairing, the system undergoes an extended-critical-localized phase transition with increasing the complex phase. The localization transition is consistent with the topological phase transition and unconventional real-complex transition of eigenenergy. In the non-Hermitian mosaic model, we provide an analytical expression of mobility edges and prove the intrinsic relation between the mobility edges and unconventional real-complex transitions. Our discoveries unveil the richness of topological and localization phenomena in non-Hermitian quasicrystals with $p$-wave pairing.