Husimi Maps in Lattices

TL;DR

This paper extends the Husimi map technique from continuous systems to lattice systems, enabling visualization of phenomena like group velocity effects, Bragg diffraction, and band scattering in lattice quantum systems.

Contribution

The paper introduces a modified Husimi map framework tailored for lattice systems, accounting for group velocity and multiple bands, and demonstrates its effectiveness in analyzing lattice-specific phenomena.

Findings

Husimi maps reveal group-velocity warping in lattices

Bragg diffraction effects are visualized using Husimi maps

Band and valley scattering points are identifiable through divergence analysis

Abstract

We build upon previous work that used coherent states as a measurement of the local phase space and extended the flux operator by adapting the Husimi projection to produce a vector field called the Husimi map. In this article, we extend its definition from continuous systems to lattices. This requires making several adjustments to incorporate effects such as group velocity and multiple bands. Several phenomena which uniquely occur in lattice systems, like group-velocity warping and internal Bragg diffraction, are explained and demonstrated using Husimi maps. We also show that scattering points between bands and valleys can be identified in the divergence of the Husimi map.