Crystallographic splitting theorem for band representations and fragile topological photonic crystals

The fundamental building blocks in band theory are band representations (BRs): 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 work aims to simplify questions on a multi-rank BR by splitting it into unit-rank bands, via the following crystallographic splitting theorem: being a rank-$N$ BR is equivalent to being splittable into a finite sum of bands indexed by $\{1,2,\ldots,N\}$, such that each band is spanned by a single, analytic Bloch function of $k$, and any symmetry in the space group acts by permuting $\{1,2,\ldots,N\}$. Applying this theorem, we develop computationally efficient methods to determine whether a given energy band (of a tight-binding or Schr\"odinger Hamiltonian) is a BR, and, if so, how to numerically construct the corresponding symmetric Wannier functions. Thus we prove that rotation-symmetric topological insulators in class AI are fragile, meaning that the obstruction to symmetric Wannier functions is removable by addition of BRs. An implication of fragility is that its boundary states, while robustly covering the bulk energy gap in finite-rank tight-binding models, are unstable 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 Yang et al. [Nature 565, 622 (2019)] is fragile topological with removable surface states, which disproves a widespread perception of 'topologically-protected' surface states in time-reversal-invariant, gapped photonic/phononic crystals. Our theorem is finally applied to derive various symmetry obstructions on the Wannier functions of topological insulators, and to prove their equivalence with the nontrivial holonomy of Bloch functions.