Two-particle collisional coordinate shifts and hydrodynamic anomalous Hall effect in systems without Lorentz invariance

TL;DR

This paper develops a gauge-invariant theory for two-particle coordinate shifts during collisions in crystals, revealing their contribution to the anomalous Hall effect in systems lacking Lorentz invariance.

Contribution

It introduces a general, gauge-invariant formula for two-particle coordinate shifts applicable to any band structure and interaction potential, linking microscopic collisions to macroscopic Hall effects.

Findings

Two-particle coordinate shifts depend on Bloch wave functions.

Shifts contribute to the anomalous Hall conductivity.

Application to Weyl fermions demonstrates the effect.

Abstract

We show that electrons undergoing a two-particle collision in a crystal experience a coordinate shift that depends on their single-particle Bloch wave functions, and derive a gauge-invariant expression for such shift, valid for arbitrary band structures, and arbitrary two-particle interaction potentials. As an application of the theory, we consider two-particle coordinate shifts for Weyl fermions in space of three spatial dimensions. We demonstrate that such shifts in general contribute to the anomalous Hall conductivity of a clean electron liquid.