Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

TL;DR

This paper develops efficient pseudospectral numerical methods for solving the time-dependent Dirac equation in static curved spaces, enabling accurate simulations of quantum particles in curved geometries with spectral convergence.

Contribution

It introduces a novel pseudospectral approach using pseudodifferential operators for the Dirac equation in curved spaces, improving computational efficiency and accuracy.

Findings

Methods achieve spectral convergence.

Effective wave absorption with absorbing layers.

Successful application to strained graphene dynamics.

Abstract

Pseudospectral numerical schemes for solving the Dirac equation in general static curved space are derived using a pseudodifferential representation of the Dirac equation along with a simple Fourier-basis technique. Owing to the presence of non-constant coefficients in the curved space Dirac equation, convolution products usually appear when the Fourier transform is performed. The strategy based on pseudodifferential operators allows for efficient computations of these convolution products by using an ordinary fast Fourier transform algorithm. The resulting numerical methods are efficient and have spectral convergence. Simultaneously, wave absorption at the boundary can be easily derived using absorbing layers to cope with some potential negative effects of periodic conditions inherent to spectral methods. The numerical schemes are first tested on simple systems to verify the convergence and are then applied to the dynamics of charge carriers in strained graphene.