Quantized electromagnetic-crystalline responses in insulators

Topological phases of matter are generally characterized by quantized responses to probe fields. In Chern insulators, magnetic fluxes can locally trap charge density, with the response coefficient being the integer-valued Chern number, i.e., the Berry curvature monopole moment in the Brillouin zone. In this work, we generalize this quantized response and investigate novel topological phases protected by nonsymmorphic symmetries in k-space, which exhibit quantized responses to both electromagnetic and crystalline gauge fields. These phases can trap charge and momentum density at magnetic and lattice defects, with coefficients given by higher-order, Z₂-valued Berry curvature multipole moments. We illustrate this with explicit lattice models that exhibit quantized Berry curvature dipole and quadrupole moments and examine their corresponding responses.