Numerical issues of the two-dimensional Dirac equation

TL;DR

This paper investigates numerical challenges in solving the two-dimensional Dirac equation, addressing fundamental issues to develop a comprehensive method validated against analytical solutions, with implications for graphene and topological insulators.

Contribution

It identifies and resolves key numerical issues in solving the 2D Dirac equation, providing a complete, validated computational method.

Findings

Developed a robust numerical method for the 2D Dirac equation.

Validated the method through comparison with analytical solutions.

Addressed fundamental numerical challenges in the equation.

Abstract

The two-dimensional Dirac equation has been widely used in graphene physics, the surface of topological insulators, and especially quantum scarring. Although a numerical approach to tackling an arbitrary confining problem was proposed several years ago, several fundamental issues must be thoroughly understood and solved. In this work, we conceal and address these challenges and finally develop a complete method, validated by comparison with analytical results.