Orbital magnetization from parallel transport of Bloch states

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.

Figures (3)

• Figure 1: Minimal model of noncoplanar anomalous Hall effect magnetFeng2020: magnetic unit cell on a 2D triangular lattice with magnetic moments on 4 sites forming a compensated noncoplanar magnetic order, shown in top-down projection (a) and from a side angle (b). The magnetic moments form a tetrahedron that remains invariant under spin-translation group symmetry operations (two-fold rotations) exchanging pairs of moments (c).
• Figure 2: The minimal model hosts four double-degenerate topological bands with nonzero Chern number (a). Both contributions to the orbital magnetization arising from the orbital magnetic moment $m^z_n$ and the Berry curvature $\Omega^{xy}_n$ show large contributions up to $1.5\,e/hc$ of opposite sign (b,c), which results into large and strongly energy-dependent orbital magnetization (d) that strongly deviate from the anomalous Hall response (e) We give the individual band contributions as dashed and dotted lines.
• Figure 3: The momentum-resolved Berry curvature $\Omega^{xy}_n(\mathbf{k})$ (a-d) and orbital magnetic moment $m^z_n(\mathbf{k})$ (e-h) in units $e/\hbar c$ for the four degenerate bands obtained within the projector formalism via the rank-2 projector given in Eq. \ref{['eqn:projectorDeg']}. We denote the BZ and the high-symmetry points and the BZ boundary (dashed lines). The color scale is normalized to the smallest and largest value for each quantity.