Filling-Enforced Gaplessness in Band Structures of the 230 Space Groups

Nonsymmorphic symmetries like screws and glides produce electron band touchings, obstructing the formation of a band insulator and leading, instead, to metals or nodal semimetals even when the number of electrons in the unit cell is an even integer. Here, we calculate the electron fillings compatible with being a band insulator for all 230 space groups, for noninteracting electrons with time-reversal symmetry. Our bounds are tight—that is, we can rigorously eliminate band insulators at any forbidden filling and produce explicit models for all allowed fillings—and stronger than those recently established for interacting systems. These results provide simple criteria that should help guide the search for topological semimetals and, also, have implications for both the nature and stability of the resulting nodal Fermi surfaces.

Figure 1

Typical band structure of a TR-symmetric free electron system with SG $\mathit{73}$. Each branch is doubly degenerate due to presence of TR and inversion. Note that the product ${\xi }_{x,\pi }^{(l)}{\xi }_{y,\pi }^{(l)}{\xi }_{z,\pi }^{(l)}$ is identical for all bands (chosen to be $+1$ here). The red dashed circle indicates inevitable crossings of four branches (a Dirac cone), enforcing $\nu =8$.