Krylov subspace methods for the Dirac equation

TL;DR

This paper evaluates the Lanczos algorithm for solving the Dirac equation, demonstrating its high precision and parallelization efficiency for large-scale relativistic quantum calculations.

Contribution

It shows that the Lanczos algorithm is effective for accurate eigenenergy computation and wave packet propagation in the Dirac equation, despite the Hamiltonian's unboundedness.

Findings

Lanczos yields precise eigenenergies and wave propagation.

Algorithm is efficiently parallelizable using MPI.

Suitable for large-scale relativistic quantum simulations.

Abstract

The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the Lanczos algorithm. As the Lanczos algorithm requires only matrix-vector products and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent parallelization capabilities are demonstrated by a parallel implementation of the Dirac Lanczos propagator utilizing the Message Passing Interface standard.