[论文解读] The Axion Insulator as a Pump of Fragile Topology

Benjamin J. Wieder, B. Andrei Bernevig|arXiv (Cornell University)|Oct 4, 2018

Dark Matter and Cosmic Phenomena被引用 74

一句话总结

本文将 axion insulator (AXI) 重新解读为一个 2D 脆弱拓扑相的循环泵，使用嵌套的 Wilson 循环将体磁电响应与角落/边缘态联系起来。

ABSTRACT

The axion insulator (AXI) has long been recognized as the simplest example of a 3D magnetic topological insulator (TI). The most familiar AXI results from magnetically gapping the surface states of a 3D $\mathbb{Z}_{2}$ TI while preserving the bulk gap. Like the 3D TI, it exhibits a quantized magnetoelectric polarizability of $θ=π$, and can be diagnosed from bulk symmetry eigenvalues when inversion symmetric. However, whereas a 3D TI is characterized by bulk Wilson loop winding, 2D surface states, and the pumping of the 2D $\mathbb{Z}_{2}$ TI index, we show that an AXI with a large number of bulk bands displays no Wilson loop winding, exhibits chiral hinge states, and does not pump any previously identified quantity. Crucially, as the AXI exhibits the topological angle $θ=π$, its occupied bands cannot be formed into maximally localized symmetric Wannier functions, despite its absence of Wilson loop winding. In this letter, we revisit the AXI from the perspective of the recently introduced notion of "fragile" topology, and discover that it in fact can be generically expressed as the cyclic pumping of a "trivialized" fragile phase: a 2D inversion-symmetric insulator with no Wilson loop winding which nevertheless carries a nontrivial topological index, the nested Berry phase $γ_{2}$. We numerically show that the nontrivial value $γ_{2}=π$ indicates the presence of anomalous 0D corner charges in a 2D insulator, and therefore, that the chiral pumping of $γ_{2}$ in a 3D AXI corresponds to the presence of chiral hinge states. We also briefly generalize our results to time-reversal-symmetric higher-order TIs, and discuss the related appearance of nontrivial $γ_{2}$ protected by $C_{2} imes\mathcal{T}$ symmetry in twisted bilayer graphene, and its implications for the presence of 0D corner states.

研究动机与目标

• 以 theta = pi 为条件动机化并表征 AXIs 为具有 θ = π 的三维磁性拓扑绝缘体。
• 研究当有大量占据带时，AXIs 是否显示 Wilson 循环缠绕与 Wannier 局域化。
• 证明 AXIs 实现了含异常角电荷的脆弱二维相的循环泵送。
• 将嵌套 Wilson 循环流与手性铰链模和磁电响应联系起来。
• 将发现推广到具时间反演对称的高阶拓扑绝缘体以及相关的 C2z x T 对称情形。

提出的方法

• 以对称于反演和时间反演的三维紧束缚模型作为父相来构建。
• 引入沿 z 方向的体铁磁势以使表面态形成能 gap，并获得 AXI 相。
• 计算 x 方向的 Wilson 循环和嵌套的 y 方向 Wilson 循环以获得 gamma1 和 gamma2。
• 证明通过添加惰性带可以将脆弱拓扑简化为平凡拓扑，从而实现嵌套 Wilson 循环分析。
• 将嵌套 Wilson 循环行列式 gamma2 与磁电角 theta 通过 theta/pi = gamma2 mod 2 联系起来。
• 在棒状和薄板几何中分析铰链和角落态，将体不变量与一维/零维边界模联系起来。

实验结果

研究问题

• RQ1 具有大量占据带的 AXI 是否显示 Wilson 循环缠绕，还是可 Wannier 化？
• RQ2 尽管不存在表面态，嵌套的 Wilson 循环是否仍能揭示 AXI 中的脆弱拓扑泵送？
• RQ3 theta = π 在 AXIs 中如何通过如 gamma2 和铰链/角落模等体不变量体现？
• RQ4 在添加平凡带以及对称性放宽（I、T，或 C2z x T）下，AXI 拓扑是否仍然稳健？

主要发现

• AXIs 可以被表述为含有异常角电荷的 2D 脆弱相的循环泵送。
• 在添加惰性/平凡带后可见的嵌套 Wilson 循环缠绕信号，指示 AXIs 的脆弱拓扑。
• theta 等于 pi 源自嵌套 Berry 相位 gamma2 的缠绕（模 2）。
• AXIs 展现出与非平凡泵送的脆弱相相一致的手性铰链模。
• 薄板在 Wilson 循环分析中表现为一个孤立的 Chern 不绝缘体，其缠绕为 C = +1。
• 该框架扩展到具有 C2z x T 对称的 AXIs，以及扭曲双层石墨烯中的相关脆弱相。