论文标题
能量带的欧拉特征数及其非全能值的原因
Euler characteristic number of the energy band and the reason for its non-integer values
论文作者
论文摘要
X. Tan等人最近在实验中观察到了我们先前工作中提出的能量谱带的拓扑特征数（参见Yu-Quan Ma等人，Arxiv：1202.2397; EPL 103，10008（2013））。莱特牧师。 \ textbf {122}，210401（2019），其中，由超导电路模拟的时间反向对称系统中的拓扑相变是被占领带的欧拉数量，而不是消失的Chern数量。但是，我们注意到，在拓扑琐事阶段，欧拉人数的非全能行为存在一些困惑。在本文中，我们表明原因很简单，因为量子度量张量$ g_ {μν} $实际上是正半明确的。在一般的二维两波段系统中，我们可以证明：（1）如果阶段是拓扑的，则量子指标必须退化（单数）〜--- $ \ det {g_ {g_ {μν}} = 0 $在第一个Brillouin区域的某个区域中。这导致了高斯 - 骨网配方的无效性，并表现出不确定的``非全能欧拉号''； （2）如果该相是拓扑不一致的，则具有非变化的浆果曲率，则量子指标将在整个第一个Brillouin区域中成为一个正确定的Riemann公制。因此，Gauss-Bonnet定理将保证能量频段的Euler数量均匀数字$χ= 2（1-G）$，并在封闭的二维Bloch Energy Band带有属属$ G $的封闭式Bloch Energy Band歧管上，这为一类非实用拓扑阶段提供了有效的拓扑指数。
The topological Euler characteristic number of the energy band proposed in our previous work (see Yu-Quan Ma et al., arXiv:1202.2397; EPL 103, 10008 (2013)) has been recently experimentally observed by X. Tan et al., Phys. Rev. Lett. \textbf{122}, 210401 (2019), in which a topological phase transition in a time-reversal-symmetric system simulated by the superconducting circuits is witnessed by the Euler number of the occupied band instead of the vanishing Chern number. However, we note that there are some confusions about the non-integer behaviors of the Euler number in the topological trivial phase. In this paper, we show that the reason is straightforward because the quantum metric tensor $g_{μν} $ is actually positive semi-definite. In a general two-dimensional two-band system, we can proved that: (1) If the phase is topological trivial, then the quantum metric must be degenerate (singular)~--- $\det {g_{μν} }=0$ in some region of the first Brillouin zone. This leads to the invalidity of the Gauss-Bonnet formula and exhibits an ill-defined ``non-integer Euler number''; (2) If the phase is topological nontrivial with a non-vanishing Berry curvature, then the quantum metric will be a positive definite Riemann metric in the entire first Brillouin zone. Therefore the Euler number of the energy band will be guaranteed an even number $χ=2(1-g)$ by the Gauss-Bonnet theorem on the closed two-dimensional Bloch energy band manifold with the genus $g$, which provides an effective topological index for a class of nontrivial topological phases.
