Communications in Mathematical Sciences

Volume 19 (2021)

Number 6

A simple real-space scheme for periodic Dirac operators

Pages: 1735 – 1750

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n6.a12

Authors

Hua Chen (Department of Physics, Colorado State University, Fort Collins, Co., U.S.A.)

Olivier Pinaud (Department of Physics, Colorado State University, Fort Collins, Co., U.S.A.)

Muhammad Tahir (Department of Physics, Colorado State University, Fort Collins, Co., U.S.A.)

Abstract

We address in this work the question of the discretization of two-dimensional periodic Dirac Hamiltonians. Standard finite difference methods on rectangular grids are plagued with the socalled Fermion doubling problem, which creates spurious unphysical modes. The classical way around the difficulty, used in the physics community is to work in the Fourier space, with the inconvenience of having to compute the Fourier decomposition of the coefficients in the Hamiltonian and related convolutions. We propose in this work a simple real-space method immune to the Fermion doubling problem and applicable to all two-dimensional periodic lattices. The method is based on spectral differentiation techniques. We apply our numerical scheme to the study of flat bands in graphene subject to periodic magnetic fields and in twisted bilayer graphene.

Keywords

Dirac equation, fermion doubling, spectral methods, flat bands

2010 Mathematics Subject Classification

35F05, 35Q41, 65M25

Full Text (PDF format)

Received 25 August 2020

Accepted 8 March 2021

Published 2 August 2021