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Dimensional reduction for anyons in the average-field approximation

Qiyun Yang

TL;DR

This work rigorously connects the two-dimensional CSS mean-field model for abelian anyons under strong anisotropic confinement to a one-dimensional quintic NLS along the loosely confined direction. By separating the tight transverse mode and a gauge-transformed ansatz, the authors derive and justify both ground-state energy reductions and time-dependent dimensional reductions, proving that the 2D ground-state energy satisfies $\lim_{\varepsilon\to 0}(E^{2D}_{\varepsilon}-e_{\varepsilon})=E^{1D}$ and that 2D dynamics converge to the 1D quintic NLS with a rate $\varepsilon^{1/4}$ under an $H^2$ well-posedness assumption. The analysis highlights a non-commutativity between the mean-field limit and dimensional reduction, and it provides a rigorous framework for connecting 2D anyon physics to quasi-1D experimental setups. Overall, the results offer a solid theoretical bridge from CSS abelian anyon dynamics in 2D to tractable 1D effective dynamics in strongly confined geometries.

Abstract

We study abelian anyons at the mean-field/almost-bosonic level, whose dynamics are governed by the Chern-Simons-Schrödinger 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 Schrödinger 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.

Dimensional reduction for anyons in the average-field approximation

TL;DR

This work rigorously connects the two-dimensional CSS mean-field model for abelian anyons under strong anisotropic confinement to a one-dimensional quintic NLS along the loosely confined direction. By separating the tight transverse mode and a gauge-transformed ansatz, the authors derive and justify both ground-state energy reductions and time-dependent dimensional reductions, proving that the 2D ground-state energy satisfies and that 2D dynamics converge to the 1D quintic NLS with a rate under an well-posedness assumption. The analysis highlights a non-commutativity between the mean-field limit and dimensional reduction, and it provides a rigorous framework for connecting 2D anyon physics to quasi-1D experimental setups. Overall, the results offer a solid theoretical bridge from CSS abelian anyon dynamics in 2D to tractable 1D effective dynamics in strongly confined geometries.

Abstract

We study abelian anyons at the mean-field/almost-bosonic level, whose dynamics are governed by the Chern-Simons-Schrödinger 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 Schrödinger equation. We rigorously establish this dimensional reduction process in the sense of ground state energies and time-dependent solutions, under an well-posedness assumption.

Paper Structure

This paper contains 14 sections, 15 theorems, 197 equations, 1 figure.

Table of Contents

  1. Introduciton
  2. Motivation
  3. Model
  4. Results
  5. Dimensional reduction for ground state energies
  6. Energy upper bound
  7. Energy lower bound
  8. Dimensional reduction for time-dependent solutions
  9. Heuristic calculations
  10. Dimensional reduction
  11. Conservative vector fields and irrotational vector fields
  12. Continuity equation from Schrödinger equations
  13. Equivalent norms for $\Sigma^s (\mathbb{R}^n)$
  14. A Strichartz estimate

Key Result

Theorem 1.2

Let $E^{2\mathrm{D}}_{\varepsilon}$ and $E^{1\mathrm{D}}_{}$ be as in Definition GSE, and let $e_{\varepsilon}$ be as in OHenergy. Then we have the relation between them as follows

Figures (1)

  • Figure 1: Exchange diagram for different limit processes on anyons.

Theorems & Definitions (31)

  • Definition 1.1: Ground state energies
  • Theorem 1.2: Ground state energies
  • Theorem 1.3: Ground states / $L^2$ normalized minimizers
  • Definition 1.4
  • Theorem 1.6: Dynamics
  • Lemma 2.1: Energy upper bound
  • proof
  • Lemma 2.2: Energy lower bound
  • Proposition 2.3: Energy decoupling
  • proof
  • ...and 21 more