Pedagogical approach to anomalous position and velocity

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.