Quantum geometry and linear orbital response in arbitrary $SU(2)$ representation

TL;DR

This work addresses computing geometric and orbital responses in multiband systems realizing arbitrary SU(2) representations, including multifold fermions. A framework based on a local quantization axis and the Wigner-Eckart theorem yields analytic expressions for the quantum geometric tensor, the orbital magnetic moment, and orbital transport coefficients across SU(2) multiplets. The method is applied to three- and four-dimensional representations and a massive spin-1 model, revealing orbital Edelstein and orbital Hall effects arising from antisymmetric band geometry components and predicting finite orbital responses in gapped systems. A key outcome is a universal decomposition of geometric quantities into a geometry-driven factor from the quantization axis and a multiplet-dependent factor from the internal SU(2) structure, indicating a common geometric origin for linear and nonlinear orbitronic phenomena.

Abstract

We develop a unified framework to compute band-geometric quantities in multiband systems whose low-energy Hamiltonians realize arbitrary $SU(2)$ representations. Exploiting the presence of a quantization axis, we use the Wigner--Eckart theorem to identify the allowed interband matrix elements and obtain compact analytic expressions for the quantum geometric tensor, the orbital magnetic moment, and the resulting orbital transport coefficients. The formalism applies to both multifold fermions and gapped $SU(2)$ models. Its versatility is demonstrated through explicit calculations in representative $SU(3)$ and $SU(4)$ settings, where orbital Edelstein and orbital Hall responses arise naturally from the antisymmetric components of the band geometry. Our results reveal a universal link between the algebraic structure of the Hamiltonian and emergent orbitronic phenomena.

Figures (1)

• Figure 1: Top panel: orbital Hall response. Bottom panel: longitudinal orbital response. Both quantities are plotted as functions of the dimensionless ratio $\xi_F=\varepsilon_F/M$ and expressed in units of $e/2\pi$. The magnitude of the orbital Hall signal is controlled by the prefactor $C_H=(e/g_L\mu_B)(\hbar v_F^2/M)$, which has been factored out of the curves to highlight the universal dependence on $\xi_F$. Similarly, the longitudinal response carries the prefactor $C_L=(e\,\tau\, v_F^2)/(8 g_L \mu_B)$, also omitted from the plots. Using realistic parameters for Dirac-like quasiparticles reported in the literature YeLinda2018KangMinguII2020KangMingu2020 ($v_F \approx 10^5\,\mathrm{m/s}$, $\tau \approx 10\,\mathrm{ps}$, $M \approx 10\,\mathrm{meV}$, and $g_L=2$), we estimate $C_H \approx 5$ and $C_L \approx 10$. These values illustrate that both orbital Hall and longitudinal orbital responses can reach experimentally relevant magnitudes in systems with moderate band velocities and small but finite gaps.