Geodesic nature and quantization of shift vector

Recently, Xu et al. introduced the concept of an interband character for a time-dependent quantum system. This quantity is gauge invariant and quantized as integer values, analogous to the Euler characteristic based on the Gauss-Bonnet theorem for a manifold with a smooth boundary. In this work, we find that the geometric shift vector in momentum space from shift currents in the bulk photovoltaic effect is equivalent to the quantum geometric potential and plays the role of geodesic curvature, that is, of a quantum system whose parameter space is the Bloch momentum. We reveal the intricate relationships among geometric quantities such as the shift vector, Berry curvature, and quantum metric. Additionally, we present the Wilson representation for the quantized interband character and extend our analysis to bosonic photon and phonon drag shift vectors with non-vertical transitions. The application of Wilson loop method facilitates first-principles calculations, providing insights into the geometric underpinnings of these interband gauge invariant quantities and shedding light on their nonlinear optical manifestations in real materials.