Topological Classification of Non-Hermitian Hamiltonians

TL;DR

This paper develops a comprehensive topological classification for non-Hermitian Hamiltonians, extending previous frameworks by incorporating homotopy theory and revealing new invariants such as braid group and cyclic groups.

Contribution

It introduces a general topological classification for non-Hermitian bands without symmetry, correcting and extending existing point-gap and line-gap schemes.

Findings

1D invariant is the noncommutative braid group, not a winding number.

2D invariants can be cyclic groups $\

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Abstract

We revisit the problem of classifying topological band structures in non-Hermitian systems. Recently, a solution has been proposed, which is based on redefining the notion of energy band gap in two different ways, leading to the so-called "point-gap" and "line-gap" schemes. However, simple Hamiltonians without band degeneracies can be constructed which correspond to neither of the two schemes. Here, we resolve this shortcoming of the existing classifications by developing the most general topological characterization of non-Hermitian bands for systems without a symmetry. Our approach, which is based on homotopy theory, makes no particular assumptions on the band gap, and predicts several amendments to the previous classification frameworks. In particular, we show that the 1D invariant is the noncommutative braid group (rather than $\mathbb{Z}$ winding number), and that depending on the braid group invariants, the 2D invariants can be cyclic groups $\mathbb{Z}_n$ (rather than $\mathbb{Z}$ Chern number). We interpret these novel results in terms of a correspondence with gapless systems, and we illustrate them in terms of analogies with other problems in band topology, namely the fragile topological invariants in Hermitian systems and the topological defects and textures of nematic liquids.