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.

中文翻译：

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无序 Bosonic Bogoliubov-de Gennes 系统的拓扑不变量

使用非对易几何方法，我们定义了无序玻色Bogoliubov-de Gennes系统的拓扑不变量，该系统具有独特的数学性质——非厄米性。为了证明定义的有效性，我们从数值上研究了二维无序人工自旋冰模型。在干净极限中，我们澄清拓扑指数与陈数完全吻合。我们还表明拓扑索引对无序具有鲁棒性。该公式提供了 magnon Hall 体系中的拓扑指数 $n_{\rm Ch}=1$ 和微不足道的局部化体系中的 $n_{\rm Ch}=0$。我们还通过示例展示了我们的方法可以扩展到其他对称类。我们的结果为进一步研究无序拓扑玻色系统铺平了道路。