# Geometric classification of non-Hermitian topological systems through the singularity ring

## TL;DR

This paper introduces a geometric classification method for non-Hermitian topological systems using a singularity ring, extending the Bloch sphere to a Bloch torus, and relates it to observable topological phenomena.

## Contribution

It generalizes the Bloch sphere to a Bloch torus for non-Hermitian systems and introduces the singularity ring as a key topological feature, providing new visualization and classification tools.

## Key findings

- The singularity ring captures degeneracy structures of exceptional points.
- Winding numbers around the SR relate to measurable Berry phases.
- The approach visualizes non-Hermitian skin effects and topological phases.

## Abstract

This work unveils how geometric features of two-band non-Hermitian Hamiltonians can completely 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, where a singularity ring (SR) captures the degeneracy structure of generic exceptional points. The 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 illustrate the richness of the interplay between SR topology and geometry beyond mere Chern number classification. Through a normalization procedure that puts generic 2-band non-Hermitian Hamiltonians on equal footing, our SR approach also allows for vivid visualization of the non-Hermitian skin effect.