Bulk-edge correspondence and stability of multiple edge states of a PI-symmetric non-Hermitian system by using non-unitary quantum walks

Topological phases and the associated multiple edge states are studied for parity and time-reversal (PT)-symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining {PT} symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the {Z} topological phase and numerically confirm that multiple edge states appear as expected from the bulk-edge correspondence. Therefore, the bulk-edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to {Z} from {Z}. In this case, we find that the number of edge states does not become one unless a pair of edge states coalesce at an exceptional point. Thereby, this is a new kind of breakdown of the bulk-edge correspondence in non-Hermitian systems. The mechanism of the prolongation of edge states against the symmetry-breaking perturbation is unique to non-Hermitian systems with multiple edge states and anti-linear symmetry. Toward experimental verifications, we propose a procedure to determine the number of multiple edge states from the time evolution of the probability distribution.