Scattering From a Two Dimensional Array of Flux Tubes: A Study of The Validity of Mean Field Theory

TL;DR

This study investigates the validity of Mean Field Theory for anyons by analyzing quantum scattering from a 2D array of magnetic flux tubes, establishing the conditions under which the approximation holds or breaks down.

Contribution

The paper develops a scattering calculation method for flux tubes arranged on a lattice and compares it to the mean field approximation, clarifying its regime of validity.

Findings

Mean Field Theory is valid when flux per tube is much less than a flux quantum.

The approximation breaks down when flux per tube approaches a quantum of flux.

The developed scattering method accurately predicts the transition from validity to breakdown.

Abstract

Mean Field Theory has been extensively used in the study of systems of anyons in two spatial dimensions. In this paper we study the physical grounds for the validity of this approximation by considering the Quantum Mechanical scattering of a charged particle from a two dimensional array of magnetic flux tubes. The flux tubes are arranged on a regular lattice which is infinitely long in the ``$y$'' direction but which has a (small) finite number of columns in the ``$x$'' direction. Their physical size is assumed to be infinitesimally small. We develop a method for computing the scattering angle as well as the reflection and transmission coefficients to lowest order in the Aharonov--Bohm interaction. The results of our calculation are compared to the scattering of the same particle from a region of constant magnetic field whose magnitude is equal to the mean field of all the flux tubes. For an incident plane wave, the Mean Field approximation is shown to be valid provided the flux in each tube is much less than a single flux quantum. This is precisely the regime in which Mean Field Theory for anyons is expected to be valid. When the flux per tube becomes of order 1, Mean Field Theory is no longer valid.