Topological classification of non-Hermitian bands

TL;DR

This paper introduces a homotopy-based topological classification framework for non-Hermitian systems, revealing new invariants related to energy level braiding and eigenstate topology, with implications for understanding non-Hermitian band structures.

Contribution

It presents a novel homotopy classification approach that uncovers additional topological invariants in non-Hermitian systems, differing from previous K-theoretical methods.

Findings

Classification decomposes into sectors based on energy braiding

Identifies torsion invariants unique to non-Hermitian systems

Shows these invariants are fragile and can be trivialized by adding bands

Abstract

We proposed a framework for the topological classification of non-Hermitian systems. Different from previous $K$-theoretical approaches, our approach is a homotopy classification, which enables us to see more topological invariants. Specifically, we considered the classification of non-Hermitian systems with separable band structures. We found that the whole classification set is decomposed into several sectors based on the braiding of energy levels and characterized by some braid group data. Each sector can be further classified based on the topology of eigenstates (wave functions). Due to the interplay between energy levels braiding and eigenstates topology, we found some torsion invariants, which only appear in the non-Hermitian world via homotopical approach. We further proved that these new topological invariants are unstable (fragile), in the sense that adding more bands will trivialize these invariants.