Wave packets in Honeycomb Structures and Two-Dimensional Dirac Equations

TL;DR

This paper investigates how wave packets near conical points in honeycomb lattice structures evolve over time, demonstrating that their dynamics are effectively described by two-dimensional Dirac equations.

Contribution

It establishes a rigorous connection between the time-evolution of wave packets in honeycomb potentials and two-dimensional Dirac equations, extending previous spectral analysis results.

Findings

Wave packets near conical points follow Dirac equation dynamics

Large-time behavior is governed by 2D Dirac equations

Validates Dirac approximation for wave evolution in honeycomb lattices

Abstract

In a recent article [10], the authors proved that the non-relativistic Schr\"odinger operator with a generic honeycomb lattice potential has conical (Dirac) points in its dispersion surfaces. These conical points occur for quasi-momenta, which are located at the vertices of the Brillouin zone, a regular hexagon. In this paper, we study the time-evolution of wave-packets, which are spectrally concentrated near such conical points. We prove that the large, but finite, time dynamics is governed by the two-dimensional Dirac equations.