Orbital Hall conductivity in a d-wave altermagnet: a multi-orbital tight-binding study

Altermagnets combine zero net magnetization with momentum-dependent spin splitting, and their hallmark transport phenomenon—the spin-splitter effect—requires no spin–orbit coupling (SOC). Here we ask how the orbital degree of freedom fits into this picture. Using a minimal multi-orbital ( \(t2g\) ) tight-binding model of a d -wave altermagnet with exact intra- \(t2g\) angular-momentum operators and the Kubo formalism, we compute on equal footing the charge, spin, and orbital Hall conductivities together with the dissipative Fermi-surface (“splitter”) spin and orbital responses. We uncover a duality between the two transport channels: in the Hall channel the orbital Hall conductivity is large and essentially independent of SOC while the spin Hall conductivity is purely SOC-induced; in the splitter channel the hierarchy is exactly reversed—the spin-splitter response is large and SOC-free, whereas its orbital counterpart vanishes identically in the SOC-free limit—a consequence of orbital-current quenching whenever the Hamiltonian admits a real matrix representation at every crystal momentum, verified explicitly for the microscopic models studied here, with the \([C_2 C4z]\) symmetry fixing the tensor structure once SOC is present—and is activated only by SOC. We verify the symmetry protection at machine precision, map the responses across the exchange–SOC parameter plane, and validate the model against spin-polarized density-functional calculations of RuO \(_2\) . Our results position d -wave altermagnets as tunable sources of orbital currents.