Crystallographic splitting theorem for band representations and fragile topological photonic crystals

Abstract

The fundamental building blocks in band theory are band representations -bands whose infinitelynumbered 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 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 Schrödinger-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 timereversal-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.CONTENTS 46 G. Tightly-bound BRs and the existence of the symmetric tight-binding limit 47 1. G-vector bundles and tight-binding lattice models 47 2. BRs and tightly-bound BRs as G-vector bundles 48 3. Existence of symmetric tight-binding limit 48 H. Lemma for Zak phases of tightly-bound band representations 48 I. Proof of localization obstruction lemma 49 References 50

Geometric approach to fragile topology beyond symmetry indicators

“…As with strong topological phases, e.g., Chern or Kane-Mele insulators, fragile topology is characterized by the obstruction to an atomic limit, while being fragile upon the coupling with an atomic insulator [35], a feature that can be directly revealed through the winding of Wilson loop [36]. Following this discovery, there has been an intense activity in the characterization of fragile topology when it is indicated by the irreducible representations of crystalline symmetries [36][37][38][39][40][41][42]. Further advances in unveiling the physical properties of such symmetry-eigenvalue-indicated fragile topology have been achieved with the prediction and observation of twisted bulkboundary correspondence [43,44].…”

Experimental characterization of fragile topology in an acoustic metamaterial

“…Moreover, many classical topological metamaterials rely on crystalline symmetries (40). This attributes fragility a more prominent role than anticipated (18,32,35). Last, the expectation that fragility plays a role in the reported strongly correlated superconductivity in twisted bilayer graphene (27)(28)(29)(30)(31)41) raises the natural question of how classical nonlinearities are influenced by these intricate band effects.…”

“…Contrarily, there are topological bands that can be trivialized by a set of bands arising from an atomic limit. In this case, our conventional notion of topological robustness is challenged, and one needs to introduce the idea of fragile topology (15)(16)(17)(18)(19)(20)(21)(22). To understand how this fragility arises, one needs to classify the bands emerging from the 230 crystalline space groups.…”

“…The first question can be answered with a definitive "no." Fragile band structures are abundant both in electronic materials ( 16), such as the flat bands of magic-angle twisted bilayer graphene (27)(28)(29)(30)(31), as well as for classical systems in photonics and phononics (18,32,33). The second question was answered in a recent complementary work (34).…”

Study on several novel topological phases in classical wave systems

“…I would like to note that in bosonic systems TRS cannot protect a QSH phase. Indeed, most analogs of QSH phase in classical wave systems have fragile topology [62][63][64]. Nevertheless, the edge states in most cases are still quite robust, as evidenced by various experiments [39, 42-44, 46, 47].…”