Pedagogical approach to anomalous position and velocity

TL;DR

This paper presents a pedagogical approach to deriving anomalous position and velocity expressions using Bloch wave functions, highlighting the roles of Berry phase and anomalous position in the context of electric field simulations.

Contribution

It introduces a novel pedagogical method that employs Bloch wave functions and optical analogies to intuitively explain anomalous velocity phenomena.

Findings

Simulation confirms transverse motion due to anomalous velocity

Additional terms from Berry phase and anomalous position are crucial

Method enhances understanding of anomalous velocity in solid-state physics

Abstract

In this work, we discuss a pedagogical method in deriving the expressions for anomalous position and velocity. While we follow the steps used in optics in the derivation of the group velocity, we use Bloch wave functions instead of plane wave states. In comparison to the plane wave case, application of Bloch wave functions results in two additional terms in the expression of the group velocity: the Berry phase factor and anomalous position contributions. These two new terms with distinct origins eventually lead to the known anomalous velocity. Aiming for an intuitive understanding, we simulate the situation under an electric field using linear-combination-of-atomic-orbital states and visually demonstrate that the envelope function exhibits the transverse motion expected from an anomalous velocity.