Abstract
We develop a graphic approach for characterizing one-dimensional non-Hermitian topological phases. The eigenstates of energy bands are mapped to a graph on the torus, where a nontrivial topology exhibits as links. The topology of band touching exceptional points is a crucial aspect of a non-Hermitian system; the existence of exceptional point results in networks. We discuss the parity-time () symmetric two-band models. The pseudo-anti-Hermiticity protects the band topology, and the eigenstate graphs in the exact -symmetric phase locate on the torus surface under the symmetry protection. For the Su-Schrieffer-Heeger ladder, the eigenstate graph is a Hopf link in the gapped nontrivial phase; chiral-time symmetry protects that the movable exceptional points appear in pairs in the real-energy gapless phase, and each exceptional point splits into a pair of exceptional points when the symmetry breaks. The proposed graphic approach is applicable in one-dimensional N-band models. Our findings provide insight into one-dimensional non-Hermitian topology phases through visualizing the eigenstates.
Highlights
The discovery of novel topological phase advances our knowledge of nature and stimulates the development of applications
In non-Hermitian topological systems, the gap closing in energy band is usually associated with exceptional point (EP) instead of degenerate points (DPs) [117,118,119, 147, 148]
In contrast to the Hermitian topological gapless phase, a non-Hermitian topological gapless phase is typically characterized by two types of winding numbers; an additional winding number solely for nonHermitian systems is defined to characterize the topology of Riemann sheet energy bands [118, 119, 142]
Summary
The discovery of novel topological phase advances our knowledge of nature and stimulates the development of applications. In non-Hermitian topological systems, the gap closing in energy band is usually associated with EPs instead of degenerate points (DPs) [117,118,119, 147, 148].
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