Fate of Majorana zero modes, exact location of critical states, and unconventional real-complex transition in non-Hermitian quasiperiodic lattices

We study a one-dimensional $p$-wave superconductor subject to non-Hermitian quasiperiodic potentials. Although non-Hermiticity exists, the Majorana zero mode is still robust against the disorder perturbation. The analytic topological phase boundary is verified by calculating the energy gap closing point and the topological invariant. Furthermore, we investigate the localized properties of this model, quantitatively revealing that the topological phase transition is accompanied by the Anderson localization phase transition, and a wide critical phase emerges with amplitude increments of the non-Hermitian quasiperiodic potentials. Finally, we numerically uncover a unconventional real-complex transition of the energy spectrum, which is different from the conventional $\mathcal{PT}$ symmetric transition.