有限和周期二维系统中局域$\mathbb{Z}_{2}$拓扑标记的理论
Theory of local $\mathbb{Z}_{2}$ topological markers for finite and periodic two-dimensional systems
AI总结：
本文提出两个空间分辨的$\mathbb{Z}_{2}$拓扑标记，用于探测有限、非均匀及无序二维时间反演对称绝缘体的局域拓扑，并在Kane-Mele模型中验证其有效性。
摘要：
二维时间反演对称绝缘体的拓扑相由$\mathbb{Z}_{2}$拓扑不变量分类。通常，该不变量通过利用时间反演对称性在倒空间中的作用来引入和计算，因此隐含地假定了周期性和均匀性。在此，我们引入两个空间分辨的$\mathbb{Z}_{2}$拓扑标记，它们能够在非均匀和有限系统的情况下探测基态电子结构的局域拓扑。第一种方法导出一个广义的局域自旋陈标记，该标记通常在自旋的垂直分量$S_{z}$不守恒时也仍然保持良好定义。第二个标记仅基于时间反演对称性，因此更为一般。我们在周期和开放边界条件下，以及在存在无序和包含拓扑/平庸异质结的情况下，用Kane-Mele模型验证了我们的标记。
英文摘要：
The topological phases of two-dimensional time-reversal symmetric insulators are classified by a $\mathbb{Z}_{2}$ topological invariant. Usually, the invariant is introduced and calculated by exploiting the way time-reversal symmetry acts in reciprocal space, hence implicitly assuming periodicity and homogeneity. Here, we introduce two space-resolved $\mathbb{Z}_{2}$ topological markers that are able to probe the local topology of the ground-state electronic structure also in the case of inhomogeneous and finite systems. The first approach leads to a generalized local spin-Chern marker, that usually remains well-defined also when the perpendicular component of the spin, $S_{z}$, is not conserved. The second marker is solely based on time-reversal symmetry, hence being more general. We validate our markers on the Kane-Mele model both in periodic and open boundary conditions, also in presence of disorder and including topological/trivial heterojunctions.
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