Bulk-boundary correspondence from the intercellular Zak phase
Bulk-boundary correspondence from the intercellular Zak phase
复制标题
DOI：
10.1103/physrevb.95.035421
复制
发表时间：
2016-08
影响因子：
3.7
通讯作者：
Jun‐Won Rhim;Jan Behrends;J. Bardarson
中科院分区：
文献类型：
--
作者：
Jun‐Won Rhim;Jan Behrends;J. Bardarson
Zak相是固体中Berry相到Bloch波函数的推广，常用于表征逆对称的一维(1D)拓扑绝缘体。然而，由于它依赖于实空间的原点和单位元胞，所以它在块状边界对应中的使用是不明确的。在这里，我们提取了元胞间ZAK相的一个与原点无关的部分，即所谓的胞间ZAK相，并证明了它是一个可以明确预测表面模数的体量。具体地说，如果存在相称的反转对称体单胞，中性有限一维紧束缚系统在费米能级以下存在${n}_{s}={\ensuremath{\gamma}}^{\mathrm{inter}}/\ensuremath{\pi}$(Mod2)带隙内表面模。我们分两步论证了这一点：首先，我们验证了$\ifmmode\pm\else\textpm\fi{}e{\ensuremath{\gamma}}^{\mathrm{inter}}/2\ensuremath{\pi}$(mod$e)$等于平移不变的一维绝缘体终止系统表面区的额外电荷积累，而剩余部分--胞内ZAK相对应于块体单位元偶极矩的电子部分。其次，我们证明了当晶胞为反对称时，额外的电荷积累与表面模数有关。我们研究了几个紧束缚模型，以定量地检验额外电荷积累与胞间Zak相的关系，以及利用胞间Zak相的体界对应关系。
The Zak phase $\ensuremath{\gamma}$, the generalization of the Berry phase to Bloch wave functions in solids, is often used to characterize inversion-symmetric one-dimensional (1D) topological insulators. Due to its dependence on the real-space origin and unit cell, however, there is an ambiguity in its use in a bulk-boundary correspondence. Here, we extract an origin-independent part of $\ensuremath{\gamma}$, the so-called intercellular Zak phase ${\ensuremath{\gamma}}^{\mathrm{inter}}$, and show that it is a bulk quantity that unambiguously predicts the number of surface modes. Specifically, a neutral finite 1D tight-binding system has ${n}_{s}={\ensuremath{\gamma}}^{\mathrm{inter}}/\ensuremath{\pi}$ (mod 2) in-gap surface modes below the Fermi level if there exists a commensurate inversion-symmetric bulk unit cell. We demonstrate this in two steps: First, we verify that $\ifmmode\pm\else\textpm\fi{}e{\ensuremath{\gamma}}^{\mathrm{inter}}/2\ensuremath{\pi}$ (mod $e)$ equals the extra charge accumulation in the surface region in a terminated system of a translationally invariant 1D insulator, while the remnant part of $\ensuremath{\gamma}$, the intracellular Zak phase ${\ensuremath{\gamma}}^{\mathrm{intra}}$, corresponds to the electronic part of the bulk's unit-cell dipole moment. Second, we show that the extra charge accumulation is related to the number of surface modes when the unit cell is inversion symmetric. We study several tight-binding models to quantitatively check both the relation between the extra charge accumulation and the intercellular Zak phase, and the bulk-boundary correspondence using the intercellular Zak phase.