Filling-enforced topological crystalline insulators

Topological crystalline insulators are topological gapped phases protected by crystal symmetries, which exhibits a variety of quantized electromagnetic phenomena. A large class of topological crystalline insulators are characterized by their non-trivial electric multipole moments, which include 1D systems with fractional polarization, and quadrupole insulators that host fractional quadrupole moments, etc.. In this work we provide theorems for the filling-enforced non-trivial topological multipole insulators. We first study a specific 1D model with 2 electrons per unit-cell protected by a spatial inversion and a glide reflection and show that it must have a fractional polarization. In the case of quadrupole insulators, we have systematically studied all the 17 wallpaper groups and established theorems that identify non-trivial quadrupole insulators enforced by certain space groups and filling conditions. Our theorems have the advantage of being robust to interactions, which serves as a powerful guide in experimental searches of non-trivial quadrupole insulators.