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

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.