A random wave model for the Aharonov-Bohm effect

TL;DR

This paper models the Aharonov-Bohm effect using random waves, deriving vortex distributions and topological charge densities influenced by magnetic flux, revealing vortex behavior near half-integer flux values.

Contribution

It introduces a novel random wave model incorporating magnetic flux, providing analytical expressions for vortex distributions related to the Aharonov-Bohm effect.

Findings

Vortex density depends on distance from flux point

Vortices are attracted to flux at half-integer values

Analytical formulas match numerical simulations

Abstract

We study an ensemble of random waves subject to the Aharonov-Bohm effect. The introduction of a point with a magnetic flux of arbitrary strength into a random wave ensemble gives a family of wavefunctions whose distribution of vortices (complex zeros) are responsible for the topological phase associated with the Aharonov-Bohm effect. Analytical expressions are found for the vortex number and topological charge densities as functions of distance from the flux point. Comparison is made with the distribution of vortices in the isotropic random wave model. The results indicate that as the flux approaches half-integer values, a vortex with the same sign as the fractional part of the flux is attracted to the flux point, merging with it at half-integer flux. Other features of the Aharonov-Bohm vortex distribution are also explored.