Avoid the Tsunami of the Dirac sea in the Imaginary Time Step method

TL;DR

This paper introduces a method to accurately compute single-particle spectra in relativistic quantum systems using the imaginary time step method, effectively avoiding the problematic diving behavior of the Dirac sea.

Contribution

The authors develop a transform-based approach that enables the ITS method to separately and reliably obtain spectra from both the Fermi and Dirac seas in relativistic equations.

Findings

The method reproduces results consistent with the conventional shooting method.

It demonstrates the feasibility and reliability of the ITS method in relativistic systems.

The approach effectively avoids the diving behavior of the Dirac sea.

Abstract

The discrete single-particle spectra in both the Fermi and Dirac sea have been calculated by the imaginary time step (ITS) method for the Schr\"{o}dinger-like equation after avoiding the "tsunami" of the Dirac sea, i.e., the diving behavior of the single-particle level into the Dirac sea in the direct application of the ITS method for the Dirac equation. It is found that by the transform from the Dirac equation to the Schr\"{o}dinger-like equation, the single-particle spectra, which extend from the positive to the negative infinity, can be separately obtained by the ITS evolution in either the Fermi sea or the Dirac sea. Identical results with those in the conventional shooting method have been obtained via the ITS evolution for the equivalent Schr\"{o}dinger-like equation, which demonstrates the feasibility, practicality and reliability of the present algorithm and dispels the doubts on the ITS method in the relativistic system.