Staggered grid leap-frog scheme for the (2+1)D Dirac equation

TL;DR

This paper introduces a staggered grid leap-frog numerical scheme for solving the (2+1)D Dirac equation, preserving dispersion, conserving norm, and handling complex electromagnetic potentials effectively.

Contribution

It presents a novel explicit staggered grid leap-frog scheme with second order accuracy for the (2+1)D Dirac equation, including stability and boundary condition implementations.

Findings

Preserves linear dispersion relation for aligned wave vectors.

Conserves the norm functional during simulations.

Effectively handles complex electromagnetic potential landscapes.

Abstract

A numerical scheme utilizing a grid which is staggered in both space and time is proposed for the numerical solution of the (2+1)D Dirac equation in presence of an external electromagnetic potential. It preserves the linear dispersion relation of the free Weyl equation for wave vectors aligned with the grid and facilitates the implementation of open (absorbing) boundary conditions via an imaginary potential term. This explicit scheme has second order accuracy in space and time. A functional for the norm is derived and shown to be conserved. Stability conditions are derived. Several numerical examples, ranging from generic to specific to textured topological insulator surfaces, demonstrate the properties of the scheme which can handle general electromagnetic potential landscapes.