Orbital physics in metals and insulators

The investigation of novel means to control the orbital degree of freedom is currently gaining momentum. Among the various effects that are under scrutiny, we focus on the orbital Hall effect, i.e., the development of a charge-neutral current, transverse to the direction of injection and carrying orbital angular momentum. To date, most theories have focused on the intra-atomic orbital degree of freedom, associated with the motion of the electron around the atom. In this work, we show that the orbital Hall effect does not only arise from intra-atomic contribution but also possesses a substantial inter-atomic contribution, associated with the gyrotropic self-rotation of the electron wavepacket. We emphasize the connection of this contribution to the Berry curvature dipole and computed both intra-atomic and inter-atomic contributions of the orbital Hall effects in selected systems. Using first-principles calculations and Kubo formula, we demonstrate that the inter-atomic orbital Hall effect can be substantial and sometimes even exceed the intra-atomic contribution. We believe that this work is of great importance in the emergent field of orbitronics as wells as in spintronics where the interplay between spin and orbital degrees of freedom plays an essential role.