Bulk–edge correspondence, spectral flow and Atiyah–Patodi–Singer theorem for the Z2-invariant in topological insulators
Bulk–edge correspondence, spectral flow and Atiyah–Patodi–Singer theorem for the Z2-invariant in topological insulators
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DOI：
10.1016/j.nuclphysb.2017.01.018
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发表时间：
2016-07
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影响因子：
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通讯作者：
Yue Yu;Yong-Shi Wu;Xin-cheng Xie
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文献类型：
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作者：
Yue Yu;Yong-Shi Wu;Xin-cheng Xie
以Fu-Kane自旋泵浦模型为例，研究了拓扑绝缘体的体边对应。我们证明了该模型中的Kane-Mele不变量是以1+1维Dirac算子族的谱流为模的Z2不变量，其整体边界条件是由系统的Kramers简并性引起的。这种谱流被定义为一个整数，它计算从负到非负的狄拉克算子族的特征值的数目和从非负到负的本征值的数目之间的差值。由于绝缘体的体态是完全有隙的，并且假设基态除了Kramers外不再简并，因此它们对谱流没有贡献，只有边缘态对谱流有贡献。Kramers数对的奇偶数与谱流的奇偶数完全相同。这揭示了边-体对应的起源，即为什么可以用边状态来表征拓扑绝缘体。此外，谱流与约化的η不变量有关，从而同时计算离散基态简并和连续的无间隙激发，这使得即使边缘态由于边缘模之间的强烈相互作用而打开了间隙，拓扑绝缘体也不同于传统的带状绝缘体。我们强调，即使对于弱无序和/或弱相互作用系统，这些结果也是有效的。更高维拓扑绝缘子的分类需要更高的谱流。
We study the bulk–edge correspondence in topological insulators by taking Fu–Kane spin pumping model as an example. We show that the Kane–Mele invariant in this model is Z 2 invariant modulo the spectral flow of a single-parameter family of 1+ 1-dimensional Dirac operators with a global boundary condition induced by the Kramers degeneracy of the system. This spectral flow is defined as an integer which counts the difference between the number of eigenvalues of the Dirac operator family that flow from negative to non-negative and the number of eigenvalues that flow from non-negative to negative. Since the bulk states of the insulator are completely gapped and the ground state is assumed being no more degenerate except the Kramers, they do not contribute to the spectral flow and only edge states contribute to. The parity of the number of the Kramers pairs of gapless edge states is exactly the same as that of the spectral flow. This reveals the origin of the edge–bulk correspondence, ie, why the edge states can be used to characterize the topological insulators. Furthermore, the spectral flow is related to the reduced η-invariant and thus counts both the discrete ground state degeneracy and the continuous gapless excitations, which distinguishes the topological insulator from the conventional band insulator even if the edge states open a gap due to a strong interaction between edge modes. We emphasize that these results are also valid even for a weak disordered and/or weak interacting system. The higher spectral flow to categorize the higher-dimensional topological insulators is expected.