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

We develop a symmetry-based framework to compute band-geometric quantities in multiband systems whose low-energy Hamiltonians realize arbitrary SU(2) representations. Exploiting the existence of a quantization axis, we use the Wigner-Eckart theorem to identify the allowed matrix elements and obtain compact analytic expressions for the quantum geometric tensor, the orbital magnetic moment, and the associated orbital transport coefficients. The formalism applies to multifold fermions as well as to gapped SU(2) models. Its versatility is illustrated through explicit calculations in representative SU(3) and SU(4) settings, where orbital Edelstein and orbital Hall responses emerge naturally from the antisymmetric components of the band geometry. These results establish a direct connection between the algebraic structure of the Hamiltonian and intrinsic orbitronic phenomena.