Homotopical classification of non-Hermitian band structures

We proposed a framework towards the topological classification of non-Hermitian band structures. Different from previous K-theoretical approaches, our approach is homotopical, 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. Each sector can be further classified based on the topology of eigenstates (wave functions). Due to the interplay between energy level braiding and eigenstates topology, we found some torsion invariants, which only appear in the non-Hermitian world. We further proved that these new topological invariants are unstable, in the sense that adding more bands will trivialize these invariants.