Manifestly gauge invariant discretizations of the Schr\"odinger equation

TL;DR

This paper develops a method to convert standard grid-based discretizations of the Schrödinger equation into gauge invariant forms, ensuring physical consistency in simulations involving magnetic fields.

Contribution

It introduces a general approach based on lattice gauge theory ideas to achieve gauge invariance in various discretization schemes, including pseudospectral methods and time integration.

Findings

Gauge invariant discretizations improve physical accuracy in magnetic field simulations.

Numerical examples show differences between gauge invariant and conventional methods.

The approach is broadly applicable to different discretization techniques.

Abstract

Grid-based discretizations of the time dependent Schr\"odinger equation coupled to an external magnetic field are converted to manifest gauge invariant discretizations. This is done using generalizations of ideas used in classical lattice gauge theory, and the process defined is applicable to a large class of discretized differential operators. In particular, popular discretizations such as pseudospectral discretizations using the fast Fourier transform can be transformed to gauge invariant schemes. Also generic gauge invariant versions of generic time integration methods are considered, enabling completely gauge invariant calculations of the time dependent Schr\"odinger equation. Numerical examples illuminating the differences between a gauge invariant discretization and conventional discretization procedures are also presented.