Topological Phases of Non-Hermitian Systems

Recent experimental advances in controlling dissipation have brought about\nunprecedented flexibility in engineering non-Hermitian Hamiltonians in open\nclassical and quantum systems. A particular interest centers on the topological\nproperties of non-Hermitian systems, which exhibit unique phases with no\nHermitian counterparts. However, no systematic understanding in analogy with\nthe periodic table of topological insulators and superconductors has been\nachieved. In this paper, we develop a coherent framework of topological phases\nof non-Hermitian systems. After elucidating the physical meaning and the\nmathematical definition of non-Hermitian topological phases, we start with\none-dimensional lattices, which exhibit topological phases with no Hermitian\ncounterparts and are found to be characterized by an integer topological\nwinding number even with no symmetry constraint, reminiscent of the quantum\nHall insulator in Hermitian systems. A system with a nonzero winding number,\nwhich is experimentally measurable from the wave-packet dynamics, is shown to\nbe robust against disorder, a phenomenon observed in the Hatano-Nelson model\nwith asymmetric hopping amplitudes. We also unveil a novel bulk-edge\ncorrespondence that features an infinite number of (quasi-)edge modes. We then\napply the K-theory to systematically classify all the non-Hermitian topological\nphases in the Altland-Zirnbauer classes in all dimensions. The obtained\nperiodic table unifies time-reversal and particle-hole symmetries, leading to\nhighly nontrivial predictions such as the absence of non-Hermitian topological\nphases in two dimensions. We provide concrete examples for all the nontrivial\nnon-Hermitian AZ classes in zero and one dimensions. In particular, we identify\na Z2 topological index for arbitrary quantum channels. Our work lays the\ncornerstone for a unified understanding of the role of topology in\nnon-Hermitian systems.\n