Visualizing one-dimensional non-hermitian topological phases

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 ($PT$) symmetric two-band models. The pseudo-anti-Hermiticity protects the band topology, and the eigenstate graphs in the exact $PT$-symmetric phase locate on the torus surface under the $PT$ 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 $PT$ 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. (paper)