Relativistic persistent currents in ideal Aharonov-Bohm rings and cylinders

TL;DR

This paper derives exact relativistic expressions for persistent currents in ideal Aharonov-Bohm rings and cylinders using the Dirac equation, revealing a saturation effect at high angular momentum and providing simple formulas for total currents.

Contribution

It presents the first exact relativistic solutions for persistent currents in Aharonov-Bohm geometries, highlighting a novel saturation effect not seen in non-relativistic models.

Findings

Relativistic partial currents relate to energy derivatives with respect to flux.

Currents saturate at high angular momentum values.

Simple formulas for total persistent currents at zero temperature are derived.

Abstract

The exact solutions of the complete (1+3)-dimensional Dirac equation of fermions moving in ideal Aharonov-Bohm (AB) rings and cylinders are used for deriving the exact expressions of the relativistic partial currents. It is shown that these currents can be related to the derivative of the fermion energy with respect to the flux parameter, just as in the non-relativistic case. However, a new and remarkable relativistic effect is the saturation of the partial currents for high values of the total angular momentum. Based on this property, the total relativistic persistent currents at $T=0$ is evaluated for rings and cylinders obtaining approximative simple closed formulas.