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Derivation of the Chern-Simons-Schrödinger equation from the dynamics of an almost-bosonic-anyon gas

Théotime Girardot, Jinyeop Lee

TL;DR

This work derives the Chern--Simons--Schrödinger (CSS) equation as the effective dynamics for an almost-bosonic extended-anyon gas in the large-$N$ limit. Using the particle-counting method of Pickl–Knowles, the authors show that, for finite times and small coupling $|eta|$, the $N$-body evolution with regularized radius $R=(\log N)^{-1/2+\varepsilon}$ remains close to a product state driven by CSS$(R,u_0)$, ultimately converging to CSS$(u_0)$ as $R\to0$ with a rate controlled by $|\,\log R|$. The analysis introduces novel controls of logarithmic divergences through Hardy-type inequalities and Grönwall-type estimates for condensate depletion and kinetic energy, enabling a rigorous link between microscopic anyon dynamics and the emergent CSS macroscopic behavior. These results provide the first derivation of CSS from a many-body anyon system, highlighting the role of the extended-anyon regularization and finite-time validity, and setting the stage for potential extensions to larger coupling or different regularizations.

Abstract

We study the time evolution of an initial product state in a system of almost-bosonic-extended-anyons in the large-particle limit. We show that the dynamics of this system can be well approximated, in finite time, by a product state evolving under the effective Chern--Simons--Schrödinger equation. Furthermore, we provide a convergence rate for the approximation in terms of the radius $R = (\log N)^{\frac{1}{2}+\varepsilon}$ of the extended anyons. These results establish a rigorous connection between the microscopic dynamics of almost-bosonic-anyon gases and the emergent macroscopic behavior described by the Chern--Simons--Schrödinger equation.

Derivation of the Chern-Simons-Schrödinger equation from the dynamics of an almost-bosonic-anyon gas

TL;DR

This work derives the Chern--Simons--Schrödinger (CSS) equation as the effective dynamics for an almost-bosonic extended-anyon gas in the large- limit. Using the particle-counting method of Pickl–Knowles, the authors show that, for finite times and small coupling , the -body evolution with regularized radius remains close to a product state driven by CSS, ultimately converging to CSS as with a rate controlled by . The analysis introduces novel controls of logarithmic divergences through Hardy-type inequalities and Grönwall-type estimates for condensate depletion and kinetic energy, enabling a rigorous link between microscopic anyon dynamics and the emergent CSS macroscopic behavior. These results provide the first derivation of CSS from a many-body anyon system, highlighting the role of the extended-anyon regularization and finite-time validity, and setting the stage for potential extensions to larger coupling or different regularizations.

Abstract

We study the time evolution of an initial product state in a system of almost-bosonic-extended-anyons in the large-particle limit. We show that the dynamics of this system can be well approximated, in finite time, by a product state evolving under the effective Chern--Simons--Schrödinger equation. Furthermore, we provide a convergence rate for the approximation in terms of the radius of the extended anyons. These results establish a rigorous connection between the microscopic dynamics of almost-bosonic-anyon gases and the emergent macroscopic behavior described by the Chern--Simons--Schrödinger equation.

Paper Structure

This paper contains 35 sections, 32 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. General Introduction
  3. The Anyon Gas
  4. Point-Like Anyons
  5. Extended Anyons
  6. The Mean-Field Limit of Almost-Bosonic Anyons
  7. The Dynamics
  8. The Hartree Dynamics
  9. The Chern--Simons--Schrödinger Equation
  10. Main Theorem
  11. Strategy of the Proof
  12. Structure of the Paper
  13. Notations and Basic Inequalities
  14. Useful Properties
  15. Operator Bounds for the Interaction Terms
  16. ...and 20 more sections

Key Result

Theorem 1.3

Let $\varphi_t$ be the solution of $\mathrm{CSS}(\varphi_0)$ defined in eq:pilotu with initial data $\varphi_0 \in H^2(\mathbb{R}^2)$. Let $\Psi_N(t)$ denote the solution of the Schrödinger equation def:schro with Hamiltonian $H_{N,R}$, where $R = (\log N)^{-\frac{1}{2}+\varepsilon}$ and $\Psi_N(0) for any choice of $\varepsilon >0$.

Theorems & Definitions (69)

  • Definition 1: Chern--Simons--Schrödinger equation
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 59 more