Conventional and unconventional anomalous velocities in multiband systems

TL;DR

This paper extends the understanding of anomalous velocities in multiband systems by deriving them beyond the single-band approximation, revealing unconventional velocities related to magnetic Bloch function singularities and confirming their derivation via different perturbation theories.

Contribution

It provides a comprehensive derivation of anomalous velocities considering multiple energy bands, highlighting the emergence of unconventional velocities and their origins.

Findings

Anomalous velocity from multiple bands differs from single-band approximation.

Unconventional anomalous velocities are perpendicular to the electric field.

Both steady-state and time-dependent perturbation theories can derive these velocities.

Abstract

The anomalous velocity has been derived so far based on the single-band approximation. In this paper, the anomalous velocity is derived accounting for multiple energy bands. It is shown that when multiple energy bands are considered, the anomalous velocity is actually derived from the velocity term which 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 which becomes zero in considering multiple energy bands. Furthermore, it is found that unconventional types of anomalous velocity may appear in addition to the 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 the uniform electric field is treated as a perturbation.