Non-Hermitian Majorana modes protect degenerate steady states

We introduce non-Hermitian generalizations of Majorana zero modes (MZMs) which appear in the topological phase of a weakly dissipative Kitaev chain coupled to a Markovian bath. Notably, the presence of MZMs ensures that the steady state in the absence of decoherence events is twofold degenerate. Within a stochastic wave-function approach, the effective Hamiltonian governing the coherent nonunitary dynamics retains BDI classification of the closed limit but belongs to one of four non-Hermitian flavors of the tenfold way. We argue for the stability of MZMs due to a generalization of particle-hole symmetry and uncover the resulting topological phase diagram. Qualitative features of our paper generalize to two-dimensional chiral superconductors. The dissipative superconducting chain can be mapped to an Ising model in a complex transverse field, and we discuss potential signatures of the degeneracy.