Edge states and topological invariants of non-Hermitian systems

Shunyu Yao, Zhong Wang

Published 2018-03-05Version 1

For Hermitian systems, the creation or annihilation of topological edge modes is accompanied by the gap closing of Bloch Hamiltonian; for non-Hermitian systems, however, the edge-state transition points can differ from the gap closing points of Bloch Hamiltonian, which indicates breakdown of the usual bulk-boundary correspondence. We study this intriguing phenomenon via exactly solving a prototype model, namely the one-dimensional non-Hermitian Su-Schrieffer-Heeger model. The solution shows that the usual Bloch waves give way to eigenstates localized at the ends of an open chain, and the Bloch Hamiltonian is not the appropriate bulk side of bulk-boundary correspondence. It is shown that the standard Brillouin zone (a unit circle for one-dimensional systems) is replaced by a deformed one (a non-unit circle for the solved model), in which topological invariants can be precisely defined, embodying an unconventional bulk-boundary correspondence. This topological invariant correctly predicts the edge-state transition points and the number of topological edge modes. The theory is of general interest to topological aspects of non-Hermitian systems.