Quantization of intra- and inter-band Berry phases in the shift current

The theory of the shift current is thus far geometrical without being topological. This means that the real-space displacement/shift of a photoexcited quasiparticle depends on the geometric Berry phase, but the Berry phase is not quantized to a rational multiple of $2\pi$. I rectify this status quo by introducing a new class of topological insulators whose band topology is \textit{only} compatible with a non-centrosymmetric space group. For such insulators, it is impossible to continuously tune the $\boldsymbol{k}$-dependent shift vector to zero throughout the Brillouin zone. Suitably averaged, the shift vector is quantized to a rational multiple of a Bravais lattice vector. Even with wide band gaps, the frequency-integrated shift conductivity greatly exceeds $e^3/h^2$, and is at least three orders of magnitude larger than the conductivity of the prototypical ferroelectric BaTiO$_3$. The large conductivity is attributed to an interplay between quantized intra- and inter-band Berry phases. In particular, topological defects of the inter-band Berry phase can enhance the shift current, even for unpolarized insulators with negligible intra-band Berry phase.