Single-cone real-space finite difference scheme for the time-dependent Dirac equation

TL;DR

This paper introduces a novel real-space finite difference scheme for the (3+1)D Dirac equation that avoids fermion doubling, conserves probability, and is highly efficient and parallelizable.

Contribution

It presents a single-cone, staggered-grid finite difference scheme for the Dirac equation that improves computational efficiency and accuracy over existing methods.

Findings

Avoids fermion doubling problem

Conserves probability density exactly

Demonstrates stability and efficiency in simulations

Abstract

A finite difference scheme for the numerical treatment of the (3+1)D Dirac equation is presented. Its staggered-grid intertwined discretization treats space and time coordinates on equal footing, thereby avoiding the notorious fermion doubling problem. This explicit scheme operates entirely in real space and leads to optimal linear scaling behavior for the computational effort per space-time grid-point. It allows for an easy and efficient parallelization. A functional for a norm on the grid is identified. It can be interpreted as probability density and is proved to be conserved by the scheme. The single-cone dispersion relation is shown and exact stability conditions are derived. Finally, a single-cone scheme for the two-component (2+1)D Dirac equation, its properties, and a simulation of scattering at a Klein step are presented.