Anderson localization and topological phase transitions in non-Hermitian Aubry-Andr\'{e}-Harper models with p-wave pairing

We study non-Hermitian Aubry-Andr\'{e}-Harper models with p-wave pairing, where the non-Hermiticity is introduced by on-site complex quasiperiodic potentials. By analysing the $\mathcal{PT}$ symmetry breaking, winding numbers of energy spectra, localization and fractal dimensions of states, and fate of Majorana fermions, a complete phase diagram on Anderson localization and topological phase transitions is obtained. In particular, the non-Hermitian topological nature of Anderson localization phase transitions from extended to critical and then to localized phases is identified, using both analytical and numerical methods. In the critical phase the complex spectrum is topological nontrivial with a fractional winding number. In the localized phase the analytical localization length of states can apply to the Hermitian case, which is absent so far. Both the non-Hermiticity and disorder are detrimental to Majorana fermions.