Quantized Crystalline-Electromagnetic Responses in Insulators

We introduce new classes of gapped topological phases characterized by quantized crystalline-electromagnetic responses, termed multipolar Chern insulators. These systems are characterized by nonsymmorphic momentum-space symmetries and mirror symmetries, leading to quantization of momentum-weighted Berry curvature multipole moments. We construct lattice models for such phases and numerically confirm their quantized responses. These systems exhibit bound charge and momentum densities at lattice and magnetic defects, as well as currents induced by electric or time-varying strain fields. Our Letter extends the classification of topological matter by uncovering novel symmetry-protected topological phases with quantized responses.