Building blocks of topological quantum chemistry: Elementary band representations

The link between chemical orbitals described by local degrees of freedom and band theory, which is defined in momentum space, was proposed by Zak several decades ago for spinless systems with and without time reversal in his theory of “elementary” band representations. In a recent paper [Bradlyn et al., Nature (London) 547, 298 (2017)] we introduced the generalization of this theory to the experimentally relevant situation of spin-orbit coupled systems with time-reversal symmetry and proved that all bands that do not transform as band representations are topological. Here we give the full details of this construction. We prove that elementary band representations are either connected as bands in the Brillouin zone and are described by localized Wannier orbitals respecting the symmetries of the lattice (including time reversal when applicable), or, if disconnected, describe topological insulators. We then show how to generate a band representation from a particular Wyckoff position and determine which Wyckoff positions generate elementary band representations for all space groups. This theory applies to spinful and spinless systems, in all dimensions, with and without time reversal. We introduce a homotopic notion of equivalence and show that it results in a finer classification of topological phases than approaches based only on the symmetry of wave functions at special points in the Brillouin zone. Utilizing a mapping of the band connectivity into a graph theory problem, we show in companion papers which Wyckoff positions can generate disconnected elementary band representations, furnishing a natural avenue for a systematic materials search.

Figure 1

Lattice basis vectors (a) and Wyckoff positions (b) of the hexagonal lattice. The (maximal) $1a, 2b$, and $3c$ Wyckoff positions are indicated by a black dot, blue squares, and red stars, respectively. The nonmaximal $6d$ and $6e$ positions are indicated by purple crosses and green squares, respectively. The multiplicity is determined by the index of the stabilizer group with respect to the point group $C_{6v}$ ($6mm$). The general position $12f$, corresponding to the orbit of a generic point, is not explicitly indicated.

Figure 2

Reciprocal lattice basis vectors and high-symmetry points of the hexagonal lattice.

Figure 3

Two possible connectivities for the EBR induced from the $2b$ position, ${\overline{\mathrm{\Gamma }}}_6\uparrow G$ with little group irreps labeled. In (a) the bands are disconnected and thus the band structure contains a group of topological bands, while in (b) they are connected. Energetics determine which phase occurs in a particular system.

Figure 4

(a) Conventional unit cell for $F222$ (SG 22). The solid blue atoms sit at the $4a$ position $(0,0,0)$ and the solid red atoms at the $4b$ position $(0,0,1/2)$. (b) Examining the $\widehat{z}$ axis shows that it is impossible to move the blue atoms to the positions of the red atoms while continuously preserving $C_{2x}$ symmetry because the symmetry requires them to move in pairs (depicted by the blue dashed circles) that transform into each other under $C_{2x}$. However, there are not enough atoms for this symmetry-preserving process to occur (or, equivalently, the dimensionality of the site-symmetry group irrep is not big enough).