Anomalous velocity

In wave mechanics, anomalous velocity refers to the group velocity of a wave packet that is transverse to an applied electric field, arising even in the absence of a magnetic field. It results from the interference of wave functions and is thus a quantum mechanical effect for the case of electrons.

When an electric field is applied to a system, electron wave packets are generally expected to move along the direction of the field. However, due to the presence of Berry curvature in momentum space, wave packets can exhibit a motion transverse to the electric field, known as anomalous motion. This phenomenon is not limited to electrons but also applies to other wave-like particles such as photons and ultracold atoms.

Related phenomena and formulation

Anomalous velocity underpins various Hall-like transport phenomena. Examples include the anomalous Hall effect in ferromagnets, the spin Hall effect in systems with significant spin-orbit coupling, and spin-dependent beam shifts in photonic systems such as the Imbert–Fedorov shift. These effects are manifestations of the topological and geometric structure of quantum states in reciprocal space, mediated by the Berry curvature, which acts as an effective magnetic field in momentum space.

The dynamics of an electron wave packet under an electric field can be described by the following equations:

${\dot {r}}_{c}=\nabla _{\vec {k}}\varepsilon ({\vec {k}})-{\dot {\vec {k}}}\times {\vec {\Omega }}({\vec {k}})$  (1)

$\hbar {\dot {\vec {k}}}=-e{\vec {E}}$  (2)

where $r_{c}$ is the center of the wave packet, $\varepsilon ({\vec {k}})$ is the band energy, and ${\vec {E}}$ is the external electric field. ${\vec {\Omega }}({\vec {k}})$ denotes the Berry curvature, defined as the curl of the Berry connection ${\vec {A}}({\vec {k}})$. The second term in the expression for ${\dot {r}}_{c}$ captures the contribution from the Berry curvature, which gives rise to the transverse anomalous motion.

Numerical visualization of anomalous velocity

The transverse group velocity (anomalous velocity) for a square lattice of $p_{x}-ip_{y}~(L=-\hbar )$ orbitals. The color represents the phase of the wave function. The phase circle around each atom due to the local angular momentum but it also overall flows to the right with a high speed (phase velocity). The dark region indicated by the white arrow is a node of the envelope function, flowing along the transverse direction. Its motion marks the anomalous velocity. Note that the anomalous velocity is much smaller than the ordinary longitudinal velocity or the phase velocity.

Visualization of group velocity can be performed by tracking the envelope function of superposed waves. A classical example from optics is the superposition of two plane waves with slightly different wave vectors, producing an envelope that moves differently from the constituent waves. However, this approach does not directly illustrate anomalous velocity. Demonstrating anomalous motion in periodic potentials requires two key modifications: (1) simulations must be performed in two or higher spatial dimensions, and (2) Bloch wave functions must be used instead of plane waves. Such simulations have been implemented using square p-orbital lattices.