  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.
