Geometric characterization of non-Hermitian topological systems through the singularity ring in pseudospin vector space

Abstract

This work unveils how geometric features of two-band non-Hermitian Hamiltonians can classify the topology of their eigenstates and energy manifolds. Our approach generalizes the Bloch sphere visualization of Hermitian systems to a ``Bloch torus'' picture for non-Hermitian systems, by extending the origin of the Bloch sphere to a singularity ring (SR) in the vector space of the real pseudospin. The SR captures the structure of generic spectral exceptional degeneracies, which emerge only if the real pseudospin vector actually falls on the SR. Applicable to non-Hermitian systems that may or may not have exceptional degeneracies, this SR picture affords convenient visualization of various symmetry constraints and reduces their topological characterization to the classification of simple intersection or winding behavior, as detailed by our explicit study of chiral, sublattice, particle-hole, and conjugated particle-hole symmetries. In 1D, the winding number about the SR corresponds to the band vorticity measurable through the Berry phase. In 2D, more complicated winding behavior leads to a variety of phases that illustrates the richness of the interplay between SR topology and geometry beyond mere Chern number classification. Through a normalization procedure that puts generic two-band non-Hermitian Hamiltonians on equal footing, our SR approach also allows for vivid visualization of the non-Hermitian skin effect.