Orbital-geometric spin response: inter-band coherence versus orbital-to-spin conversion

Orbitronics seeks to convert orbital angular momentum (OAM) into spin, but its microscopic dynamics is not understood. Here I show that, at moderate spin-orbit coupling, an electrically induced spin density contains a direct term and an $\mathit{\text{orbital-geometric}}\mathit{\text{spin}}\mathit{\text{response}}$ (OGSR) due to inter-band coherence of the spinless parent Hamiltonian—related to OAM but not descended from it. In massive Dirac fermions the OGSR survives as the net OAM vanishes, and generates $\mathit{\text{counterflow}}$, opposing the direct response. These results provide guidelines for optimising orbitronic systems.