Geodesic nature and quantization of shift vector

TL;DR

This paper explores the geometric and quantized properties of the shift vector in quantum systems, linking it to topological invariants and optical responses, using the Wilson loop approach for first-principles calculations.

Contribution

It introduces the quantization of the gauge-invariant shift vector and its geometric relationships, extending analysis to phonon drag shift vectors with non-vertical transitions.

Findings

Shift vector can be quantized as integer values.

Loop integral of the shift vector affects nonlinear optical responses.

Wilson loop method enables first-principles calculations of geometric quantities.

Abstract

We present the geodesic nature and quantization of geometric shift vector in quantum systems, with the parameter space defined by the Bloch momentum, using the Wilson loop approach. Our analysis extends to include bosonic phonon drag shift vectors with non-vertical transitions. We demonstrate that the gauge invariant shift vector can be quantized as integer values, analogous to the Euler characteristic based on the Gauss-Bonnet theorem for a manifold with a smooth boundary. We reveal intricate relationships among geometric quantities such as the shift vector, Berry curvature, and quantum metric. Our findings demonstrate that the loop integral of the shift vector in the quantized interband formula contributes to the non-quantized component of the trace of conductivity in the circular photogalvanic effect. The Wilson loop method facilitates first-principles calculations, providing insights in the geometric underpinnings of these interband gauge invariant quantities and shedding light on their nonlinear optical manifestations in real materials.