Defective Majorana zero modes in non-Hermitian Kitaev chain

We investigate the number-anomalous of the Majorana zero modes in the non-Hermitian Kitaev chain, whose hopping and superconductor paring strength are both imbalanced. We find that the combination of two imbalanced non-Hermitian terms can induce defective Majorana edge states, which means one of the two localized edge states will disappear due to the non-Hermitian suppression effect. As a result, the conventional bulk-boundary correspondence is broken down. Besides, the defective edge states are mapped to the ground states of non-Hermitian transverse field Ising model, and the global phase diagrams of ferromagnetic-antiferromagnetic crossover for ground states are given. Our work, for the first time, reveal the break of topological robustness for the Majorana zero modes, which predict more novel effects both in topological material and in non-Hermitian physics.