Orbital magnetization from parallel transport of Bloch states

TL;DR

This paper introduces a new geometric operator for quantum states that enables a more general and comprehensive calculation of orbital magnetization in materials, including cases with degenerate bands.

Contribution

It presents a band-resolved adiabatic connection operator as a novel tool for quantum geometry, extending orbital magnetization calculations to degenerate bands.

Findings

Formulated a gauge-invariant expression for orbital magnetic moments.

Demonstrated the theory on a noncoplanar anomalous Hall magnet.

Removed limitations of previous Bloch-state formulas for degenerate bands.

Abstract

Quantum geometric formulations of linear and nonlinear responses can be constructed from a single building block in the form of a gauge-invariant interband transition operator. Here, we identify a second building block for quantum geometry: a band-resolved adiabatic connection operator that captures the noncommutativity between band projectors and their momentum derivatives. The band-resolved adiabatic connection operator, first introduced in the theory of adiabatic driving, serves as a generalized angular momentum within the state manifold of single bands, and we employ it to reformulate expressions for the band-resolved orbital magnetic moment. This form provides a complementary geometric interpretation alongside the multiband separation between energetic- and quantum-state properties by the two-state Berry curvature. Our formalism allows us to present formulas valid for both nondegenerate and degenerate bands, thereby removing the limitations of the common Bloch-state formula. We illustrate our theory by calculating a large orbital magnetization emerging without spin-orbit coupling in a spin-compensated, noncoplanar anomalous Hall magnet with degenerate bands.