Resolving the spurious-state problem in the Dirac equation with finite difference method

TL;DR

This paper presents a simple and efficient numerical method to solve the Dirac equation using finite difference techniques, effectively eliminating the spurious states caused by fermion doubling.

Contribution

The authors introduce an asymmetric difference formula combined with parity-based alternation to resolve the spurious-state problem in finite difference solutions of the Dirac equation.

Findings

Elimination of spurious states in Dirac equation solutions.

Improved numerical stability and accuracy.

Applicability to various relativistic problems.

Abstract

To solve the Dirac equation with the finite difference method, one has to face up to the spurious-state problem due to the fermion doubling problem when using the conventional central difference formula to calculate the first-order derivative on the equal interval lattices. This problem is resolved by replacing the central difference formula with the asymmetric difference formula, i.e., the backward or forward difference formula. To guarantee the hermitian of the Hamiltonian matrix, the backward and forward difference formula should be used alternatively according to the parity of the wavefunction. This provides a simple and efficient numerical prescription to solve various relativistic problems in the microscopic world.