Dimensional reduction for anyons in the average-field approximation

TL;DR

This paper rigorously derives a one-dimensional effective model for abelian anyons in a mean-field setting, showing that under anisotropic trapping, the complex 2D dynamics reduce to a quintic nonlinear Schr"odinger equation.

Contribution

It introduces a novel dimensional reduction technique for the Chern-Simons-Schr"odinger system, connecting 2D anyon dynamics to a 1D nonlinear Schr"odinger equation.

Findings

Effective 1D model derived from 2D system

Rigorous proof of dimensional reduction

Convergence in ground state energies and solutions

Abstract

We study abelian anyons at the mean-field/almost-bosonic level, whose dynamics are governed by the Chern-Simons-Schr\"odinger system. We consider the dimensional reduction of this 2D model by introducing an anisotropic trapping potential, and derive an effective 1D model after tracing out the tight confinement direction. The resulting effective dynamics in the loose confinement direction is captured by a quintic defocusing nonlinear Schr\"odinger equation. We rigorously establish this dimensional reduction process in the sense of ground state energies and time-dependent solutions, under an $H^2$ well-posedness assumption.