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Dimensional reduction for a system of 2D anyons

Nicolas Rougerie, Qiyun Yang

Abstract

Anyons with a statistical phase parameter $α\in(0,2)$ are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave-guide. More specifically, we study the 2D magnetic-gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the $y$-direction than in the $x$-direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the $x$-direction is given by the impenetrable Tonks-Girardeau Bose gas, which has no dependency on $α$, and no trace left of the long-range interactions of the 2D model.

Dimensional reduction for a system of 2D anyons

Abstract

Anyons with a statistical phase parameter are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave-guide. More specifically, we study the 2D magnetic-gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the -direction than in the -direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the -direction is given by the impenetrable Tonks-Girardeau Bose gas, which has no dependency on , and no trace left of the long-range interactions of the 2D model.
Paper Structure (15 sections, 21 theorems, 215 equations)

This paper contains 15 sections, 21 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The model
  4. Main results and discussion
  5. Preliminaries
  6. Domain of the Hamiltonians
  7. 2D model
  8. 1D model
  9. Hardy inequalities for 2D anyons
  10. Eigenvalues and eigenfunctions
  11. 1D model
  12. 2D model
  13. Energy upper bounds
  14. Energy lower bounds
  15. Relation between eigenfunctions

Key Result

Theorem 1.1

Let $\lambda^{\mathrm{2D}}_{k}$ and $\lambda^{1\mathrm{D}}_k$ be the $k$-th eigenvalues of $H^{\mathrm{2D}}_{\varepsilon}$ and $H^{1\mathrm{D}}$ respectively (counting the multiplicity), and let $e_{\varepsilon}$ be the ground energy of $H^{\mathrm{HO}}_{\varepsilon}$. For any fixed $k$, we have, in

Theorems & Definitions (38)

  • Theorem 1.1: Relation between energies
  • Theorem 1.2: Relation between eigenfunctions
  • Theorem 2.1: Isolated two-anyon Hardy
  • proof
  • Lemma 2.2: Three-particle Hardy
  • Remark 2.3: Three-particle Hardy for anyons
  • Theorem 2.4: Many-anyon Hardy for $\alpha \in (0,2)$
  • proof
  • Corollary 2.5: $H^1$-regularity
  • proof
  • ...and 28 more