Topological Invariant for Bosonic Bogoliubov-de Gennes Systems with Disorder

Using the method of noncommutative geometry, we define a topological invariant in disordered bosonic Bogoliubov-de Gennes systems, which possess a unique mathematical property---non-Hermiticity. To demonstrate the validity of the definition, we investigate a disordered artificial spin ice model in two dimensions numerically. In the clean limit, we clarify that the topological index perfectly coincides with the Chern number. We also show that the topological index is robust against disorder. The formula provides the topological index $n_{\rm Ch}=1$ in the magnon Hall regime and $n_{\rm Ch}=0$ in a trivial localized one. We also show by example that our method can be extended to other symmetry classes. Our results pave the way for further studies on topological bosonic systems with disorder.