Homotopical characterization of non-Hermitian band structures

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