A dispersion and norm preserving finite difference scheme with transparent boundary conditions for the Dirac equation in (1+1)D

TL;DR

This paper introduces a finite difference scheme for the (1+1)D Dirac equation that preserves dispersion and norm, employs exact transparent boundary conditions, and effectively simulates wave packet dynamics without boundary reflections.

Contribution

It develops a novel leap-frog finite difference scheme with exact discrete transparent boundary conditions for the Dirac equation, avoiding fermion doubling and preserving key physical properties.

Findings

Accurately simulates wave packets leaving the domain without reflection.

Preserves the dispersion relation exactly for massless case.

Demonstrates stability and effectiveness of the boundary conditions.

Abstract

A finite difference scheme is presented for the Dirac equation in (1+1)D. It can handle space- and time-dependent mass and potential terms and utilizes exact discrete transparent boundary conditions (DTBCs). Based on a space- and time-staggered leap-frog scheme it avoids fermion doubling and preserves the dispersion relation of the continuum problem for mass zero (Weyl equation) exactly. Considering boundary regions, each with a constant mass and potential term, the associated DTBCs are derived by first applying this finite difference scheme and then using the Z-transform in the discrete time variable. The resulting constant coefficient difference equation in space can be solved exactly on each of the two semi-infinite exterior domains. Admitting only solutions in $l_2$ which vanish at infinity is equivalent to imposing outgoing boundary conditions. An inverse Z-transformation leads to exact DTBCs in form of a convolution in discrete time which suppress spurious reflections at the boundaries and enforce stability of the whole space-time scheme. An exactly preserved functional for the norm of the Dirac spinor on the staggered grid is presented. Simulations of Gaussian wave packets, leaving the computational domain without reflection, demonstrate the quality of the DTBCs numerically, as well as the importance of a faithful representation of the energy-momentum dispersion relation on a grid.