Crystallographic splitting theorem for band representations and fragile topological photonic crystals
Crystallographic splitting theorem for band representations and fragile topological photonic crystals
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DOI：
10.1103/physrevb.102.115117
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发表时间：
2020-09-09
影响因子：
3.7
通讯作者：
Lu, Ling
中科院分区：
文献类型：
--
作者：
Alexandradinata, A.;Holler, J.;Lu, Ling
带理论中的基本构件是带表示-其无穷多个Wannier函数(通过空间群的作用)由以空间中的一点为中心的有限数量的对称Wannier函数生成的带。本文旨在通过以下晶体分裂定理将多秩带表示的问题简化为单位秩带表示：作为秩N带表示等价于可分裂成由{1，2，…，N}索引的有限个带之和，使得每个带由k的单个解析Bloch函数跨越，并且空间群中的任何对称性通过置换{1，2，...，N}而起作用。对于Wannier函数在整数自旋表示下变换的所有晶体空间群的能带表示，我们证明了这个定理；在半整数自旋的情况下，该定理的唯一例外是具有立方点群的三维空间空间群。应用这一定理，我们发展了计算有效的方法来确定给定的(紧束缚或薛定谔哈密顿量)能带是否是带表示，如果是，如何数值构造相应的对称Wannier函数。从而证明了Wigner-Dyson类AI中的旋转对称拓扑绝缘子是脆弱的，这意味着可以通过在填充带子空间中加入带表示来消除对对称Wannier函数的阻碍。脆弱性的一个含义是，如果希尔伯特空间扩展到包括所有对称允许的表示，那么它的边界态虽然强健地覆盖了有限秩紧束缚模型中的体能隙，但可以不稳定。这些易碎的绝缘体有我们识别的光子类似物；特别是，我们证明了由杨一浩等人建造的现有光子晶体。[Natural 565,622(2019)]是一种具有可移除边界态的脆弱拓扑结构，它驳斥了人们普遍认为的在时间反转不变、有间隙的光子/声子晶体中“拓扑保护”的边界态的看法。作为定理的最后应用，我们导出了拓扑绝缘子的Wannier函数上的各种对称阻挡；对于某些空间群，证明了这些阻挡等价于Bloch函数的非平凡完整。
The fundamental building blocks in band theory are band representations-bands whose infinitely numbered Wannier functions are generated (by action of a space group) from a finite number of symmetric Wannier functions centered on a point in space. This paper aims to simplify questions on a multirank band representation by splitting it into unit-rank bands via the following crystallographic splitting theorem: Being a rank-N band representation is equivalent to being splittable into a finite sum of bands indexed by {1, 2, ..., N}, such that each band is spanned by a single, analytic Bloch function of k, and any symmetry in the space group acts by permuting {1, 2, ..., N}. We prove this theorem for all band representations (of crystallographic space groups) whose Wannier functions transform in the integer-spin representation; in the half-integer-spin case, the only exceptions to the theorem exist for three-spatial-dimensional space groups with cubic point groups. Applying this theorem, we develop computationally efficient methods to determine whether a given energy band (of a tight-binding or Schrodinger-type Hamiltonian) is a band representation and, if so, how to numerically construct the corresponding symmetric Wannier functions. Thus we prove that rotation-symmetric topological insulators in Wigner-Dyson class AI are fragile, meaning that the obstruction to symmetric Wannier functions can be removed by addition of band representations to the filled-band subspace. An implication of fragility is that its boundary states, while robustly covering the bulk energy gap in finite-rank tight-binding models, can be destabilized if the Hilbert space is expanded to include all symmetry-allowed representations. These fragile insulators have photonic analogs that we identify; in particular, we prove that an existing photonic crystal built by Yihao Yang et al. [Nature 565, 622 (2019)] is fragile topological with removable boundary states, which disproves a widespread perception of "topologically protected" boundary states in time-reversal-invariant, gapped photonic/phononic crystals. As a final application of our theorem, we derive various symmetry obstructions on the Wannier functions of topological insulators; for certain space groups, these obstructions are proven to be equivalent to the nontrivial holonomy of Bloch functions.