Solving Dirac equations on a 3D lattice with inverse Hamiltonian and spectral methods

TL;DR

This paper introduces a novel 3D lattice method for solving the Dirac equation that avoids variational collapse and fermion doubling, demonstrating high accuracy across various potential deformations.

Contribution

The paper presents a new spectral method combining inverse Hamiltonian and Fourier transform techniques for accurate Dirac equation solutions on a 3D lattice.

Findings

Differences in single particle energy less than 10^{-4} MeV compared to shooting method

Densities are nearly identical to benchmark results

Applicable to various deformed potentials without modifications

Abstract

A new method to solve the Dirac equation on a 3D lattice is proposed, in which the variational collapse problem is avoided by the inverse Hamiltonian method and the fermion doubling problem is avoided by performing spatial derivatives in momentum space with the help of the discrete Fourier transform, i.e., the spectral method. This method is demonstrated in solving the Dirac equation for a given spherical potential in 3D lattice space. In comparison with the results obtained by the shooting method, the differences in single particle energy are smaller than $10^{-4}$~MeV, and the densities are almost identical, which demonstrates the high accuracy of the present method. The results obtained by applying this method without any modification to solve the Dirac equations for an axial deformed, non-axial deformed, and octupole deformed potential are provided and discussed.