Quantized shift response in multi-gap topological phases

We show that certain 3D multi-gap topological insulators can host quantized shift photoconductivities due to bulk invariants that are defined under reality conditions imposed by additional symmetries. We recast the quantization in terms of the integrated torsion tensor and the non-Abelian Berry connection constituting Chern-Simons forms. Physically, we recognize that the topological quantization emerges purely from virtual transitions contributing to the optical response. Our findings provide another quantized electromagnetic DC response due to the non-trivial band topology, beyond the quantum anomalous Hall effect of Chern insulators and quantized circular photogalvanic effect found in Weyl semimetals.