Dimensional reduction on a sphere

TL;DR

This paper explores the dimensional reduction of 2D quantum models on a sphere to 1D models on a circle, aiming to connect 2D anyon models with 1D Calogero-Sutherland models for better understanding of their statistical properties.

Contribution

It introduces a new spherical anyon-like model derived from the Aharonov-Bohm problem to facilitate dimensional reduction from 2D to 1D.

Findings

Mapping from 2D sphere to 1D circle established for free particles.

Standard anyon model on the sphere is inadequate for reduction.

A new spherical anyon-like model is proposed based on the Aharonov-Bohm effect.

Abstract

The question of the dimensional reduction of two-dimensional (2d) quantum models on a sphere to one-dimensional (1d) models on a circle is adressed. A possible application is to look at a relation between the 2d anyon model and the 1d Calogero-Sutherland model, which would allow for a better understanding of the connection between 2d anyon exchange statistics and Haldane exclusion statistics. The latter is realized microscopically in the 2d LLL anyon model and in the 1d Calogero model. In a harmonic well of strength \omega or on a circle of radius R - both parameters \omega and R have to be viewed as long distance regulators - the Calogero spectrum is discrete. It is well known that by confining the anyon model in a 2d harmonic well and projecting it on a particular basis of the harmonic well eigenstates, one obtains the Calogero-Moser model. It is then natural to consider the anyon model on a sphere of radius R and look for a possible dimensional reduction to the Calogero-Sutherland model on a circle of the same radius. First, the free one-body case is considered, where a mapping from the 2d sphere to the 1d chiral circle is established by projection on a special class of spherical harmonics. Second, the N-body interacting anyon model is considered : it happens that the standard anyon model on the sphere is not adequate for dimensional reduction. One is thus lead to define a new spherical anyon-like model deduced from the Aharonov-Bohm problem on the sphere where each flux line pierces the sphere at one point and exits it at its antipode.