Conventional and unconventional anomalous velocities in multiband systems

Anomalous velocity has been derived so far based on the single-band approximation. In this paper, anomalous velocity is derived taking into account multiple energy bands. It is shown that when multiple energy bands are considered, anomalous velocity is actually derived from the velocity term that goes to zero under the single-band approximation. It is also shown that the anomalous velocity based on the single-band approximation is derived improperly from the velocity term that becomes zero in considering multiple energy bands. Furthermore, it is found that unconventional types of anomalous velocity may appear in addition to conventional anomalous velocity. These unconventional anomalous velocities are perpendicular to the electric field and come from the singularity of the magnetic Bloch function in the magnetic Brillouin zone. It is confirmed that conventional and unconventional anomalous velocities can also be derived not only from the steady-state perturbation theory, but also from the time-dependent perturbation theory in which a time-dependent vector potential yielding a uniform electric field is treated as a perturbation.