Abstract
Recent experimental advances in controlling dissipation have brought about unprecedented flexibility in engineering non-Hermitian Hamiltonians in open classical and quantum systems. A particular interest centers on the topological properties of non-Hermitian systems, which exhibit unique phases with no Hermitian counterparts. However, no systematic understanding in analogy with the periodic table of topological insulators and superconductors has been achieved. In this paper, we develop a coherent framework of topological phases of non-Hermitian systems. After elucidating the physical meaning and the mathematical definition of non-Hermitian topological phases, we start with one-dimensional lattices, which exhibit topological phases with no Hermitian counterparts and are found to be characterized by an integer topological winding number even with no symmetry constraint, reminiscent of the quantum Hall insulator in Hermitian systems. A system with a nonzero winding number, which is experimentally measurable from the wave-packet dynamics, is shown to be robust against disorder, a phenomenon observed in the Hatano-Nelson model with asymmetric hopping amplitudes. We also unveil a novel bulk-edge correspondence that features an infinite number of (quasi-)edge modes. We then apply the K-theory to systematically classify all the non-Hermitian topological phases in the Altland-Zirnbauer classes in all dimensions. The obtained periodic table unifies time-reversal and particle-hole symmetries, leading to highly nontrivial predictions such as the absence of non-Hermitian topological phases in two dimensions. We provide concrete examples for all the nontrivial non-Hermitian AZ classes in zero and one dimensions. In particular, we identify a Z2 topological index for arbitrary quantum channels. Our work lays the cornerstone for a unified understanding of the role of topology in non-Hermitian systems.
Highlights
For the sake of comparison with SPT phases in Hermitian systems, we focus primarily on lattice systems described by non-Hermitian Bloch Hamiltonians HðkÞ, but our formalism can be applied to other setups like quantum channels [112] and full counting statistics [107], where non-Hermiticity appears in completely positive trace-preserving (CPTP) superoperators and generators for characteristic functions, respectively
On the basis of these two guiding principles, we find that a one-dimensional lattice with asymmetric hopping amplitudes turns out to be the most prototypical example comparable to the quantum-Hall insulator, in the sense that an integer topological number can be defined without any symmetry protection
The two guiding principles are a dynamical viewpoint on topological systems and the constraint such that the energy spectrum neither touches nor crosses the base point
Summary
Topological phases of matter [1,2,3,4,5] have attracted growing interest over the past decade in many subfields of physics, including condensed-matter physics [6,7,8,9,10,11,12], ultracold atomic gases [13,14,15,16,17,18,19,20,21], quantum information [22,23,24,25], photonics [26,27,28,29,30,31,32,33,34,35], and mechanics [36,37,38,39]. On the basis of these two guiding principles, we find that a one-dimensional lattice with asymmetric hopping amplitudes turns out to be the most prototypical example comparable to the quantum-Hall insulator, in the sense that an integer topological number can be defined without any symmetry protection This result gives an interesting topological interpretation to the emergent Anderson transition [113] in the Hatano-Nelson model [114,115,116], which should otherwise be absent in one-dimensional Hermitian systems [117]. Several technical details and an experimental implementation on asymmetric hopping are relegated to Appendixes to avoid digressing from the main subjects
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