Crystallographic splitting theorem for band representations and fragile topological photonic crystals

Article

关键词: ENERGY-BANDS;    WANNIER FUNCTIONS;    BLOCH-BUNDLES;    SYMMETRY;    INSULATOR;    PHASE;    ELECTRONS;    LATTICE;    STATES;    WAVE;   

DOI  :  10.1103/PhysRevB.102.115117

来源: SCIE

【 摘 要 】

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.

【 授权许可】

Free