The multi-state geometry of shift current and polarization

TL;DR

This paper introduces a gauge-invariant formalism based on quantum state projectors to better understand multi-band Bloch state properties, with applications to non-linear optics and electronic polarization.

Contribution

It develops a new projector-based approach to analyze multi-state geometry, clarifying the relation between shift current and polarization, and demonstrating its application to TMDs and model systems.

Findings

Derived a gauge-invariant expression for shift current.

Applied the formalism to transition metal dichalcogenides.

Analyzed contributions of multi-state geometry to optical properties.

Abstract

The quantum metric and Berry curvature capture essential properties of non-trivial Bloch states and underpin many fascinating phenomena. However, it becomes increasingly evident that a more comprehensive understanding of quantum state geometry is necessary to explain properties involving Bloch states of multiple bands, such as optical transitions. To this end, we employ quantum state projectors to develop an explicitly gauge-invariant formalism and demonstrate its power with applications to non-linear optics and the theory of electronic polarization. We provide a simple expression for the shift current that resolves its precise relation to the moments of electronic polarization, clarifies the treatment of band degeneracies, and reveals its decomposition into the sum of the skewness of the occupied states and intrinsically multi-state geometry. The projector approach is applied to calculate non-linear optical properties of transition metal dichalcogenides (TMDs) layers, using previously calculated minimal tight-binding models, and demonstrated analytically on a three-band generalization of the Rice-Mele chain to elucidate the different contributions. We close with comments on further applications of the projector operator approach to multi-state geometry.