A simple real-space scheme for periodic Dirac operators

TL;DR

This paper introduces a straightforward real-space spectral differentiation method for discretizing 2D periodic Dirac Hamiltonians, effectively avoiding Fermion doubling issues and applicable to various lattice structures, demonstrated on graphene systems.

Contribution

A novel real-space spectral differentiation scheme that overcomes Fermion doubling in discretizing 2D periodic Dirac operators, applicable to all lattice types.

Findings

Successfully applied to flat band studies in graphene.

Avoids Fermion doubling without Fourier transforms.

Applicable to all 2D periodic lattices.

Abstract

We address in this work the question of the discretization of two-dimensional periodic Dirac Hamiltonians. Standard finite differences methods on rectangular grids are plagued with the so-called 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.