Systematic study for two-dimensional Z2 topological phase transitions at high-symmetry points in all layer groups
Systematic study for two-dimensional Z2 topological phase transitions at high-symmetry points in all layer groups
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DOI：
10.1103/physrevb.107.115120
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发表时间：
2022-10
影响因子：
3.7
通讯作者：
R. Sasaki;Yutaro Tanaka;S. Murakami
中科院分区：
文献类型：
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作者：
R. Sasaki;Yutaro Tanaka;S. Murakami
我们构建了具有时间反转对称性的二维系统中 $Z_2$ 拓扑相变的一般理论。我们研究了所有 80 层组中 $k$ 空间中所有高对称点处能带反转时 $Z_2$ 拓扑相变的可能性。我们排除具有反演对称性的层组，因为已知 $Z_2$ 拓扑相变与奇偶校验交换的能带反转相关。在其他层组中，我们发现 21 个具有能带反转的绝缘体到绝缘体跃迁的层组，这个问题最终简化为五个点组 $C_3、C_4、C_6、S_4$ 和 $C_{3h}$。我们展示了能带反转时 $Z_2$ 拓扑不变量的变化如何完全由高对称点处占据和未占据能带的不可表示决定。例如，在 $C_3$ 的情况下，我们表明只要 $C_3$ 特征值为 $\{e^{\pi i / 3}、e^{-\pi i / 3}\}$ 和 $\{-1, -1\}$ 的两个 Kramer 对之间发生能带反转，$Z_2$ 拓扑不变量就会发生变化。这些结果不包含在基于对称性的指示剂理论或拓扑量子化学中。
We construct a general theory of $Z_2$ topological phase transitions in two-dimensional systems with time-reversal symmetry. We investigate the possibilities of $Z_2$ topological phase transitions at band inversions at all high-symmetry points in $k$-space in all the 80 layer groups. We exclude the layer groups with inversion symmetry because the $Z_2$ topological phase transition is known to be associated with band inversions with an exchange of parities. Among the other layer groups, we find 21 layer groups with insulator-to-insulator transitions with band inversion, and this problem is finally reduced to five point groups $C_3, C_4, C_6, S_4$, and $C_{3h}$. We show how the change of the $Z_2$ topological invariant at a band inversion is entirely determined by the irreps of occupied and unoccupied bands at the high-symmetry point. For example, in the case of $C_3$, we show that the $Z_2$ topological invariants change whenever the band inversion occurs between two Kramers pairs whose $C_3$ eigenvalues are $\{e^{\pi i / 3}, e^{-\pi i / 3}\}$ and $\{-1, -1\}$. These results are not included in the theory of symmetry-based indicators or topological quantum chemistry.