Topological Invariant for Bosonic Bogoliubov-de Gennes Systems with Disorder*

The highly successful studies on the topological phases for fermions have been extended to bosonic systems (see, e.g., a review [1]). As a difference from fermionic systems, bosonic Bogoliubov-de Gennes systems possess a unique mathematical property---non-Hermiticity. In addition, since topological invariants that characterize the topological phases are usually defined in terms of Bloch wave functions in crystal momentum space, the determination of the invariants for disordered systems is still very challenging.
In this study, we define a topological invariant corresponding to the conventional Chern number in disordered magnon Hall systems, using the method of non-commutative geometry. To demonstrate the validity of the definition, we study a disordered artificial spin ice model in two dimensions numerically. In the clean limit of the model, we clarify that the topological index perfectly coincides with the Chern number. We also show that the topological index is robust against disorder, and the index characterizes the disordered topological and trivial localized magnon phases as we map out the phase diagram [2].
[1] H. Kondo, Y. Akagi, and H. Katsura, Preprint arXiv:2006.10391 (2020).
[2] Y. Akagi, Preprint arXiv:2010.07762 (2020).