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Microscopic derivation of the stationary Chern-Simons-Schrödinger equation for almost-bosonic anyons

Alireza Ataei, Douglas Lundholm, Théotime Girardot

TL;DR

The paper rigorously derives a stationary Chern-Simons-Schrödinger (CSS) energy functional as the effective description of a two-dimensional quantum gas of almost-bosonic anyons with flux attachment and spin-orbit coupling. By introducing a Hartree-Jastrow trial state and a precise scaling limit where the anyon radius R_N→0 and the anyon-flux parameter α_N scales as α_N=β/(N−1), the authors show that, for β_N=(N−1)α_N→β>0 and ω_N→ω∈[0,∞], the energy per particle converges to the CSS energy density ∫|(-i∇+βA[|u|^2])u|^2 + ∫V|u|^2 + 2πβ G(2βω,g)∫|u|^4, with the one-particle density tending to |u|^2. The key technical achievement is a rigorous construction of the two-body Jastrow factor f and meticulous control of kinetic, potential, singular, and current terms to obtain the exact leading-order contribution, including the scale-dependent function G that encodes the two-body physics and the spin-orbit coupling g. The results connect microscopic parameters to macroscopic, self-consistent magnetic interactions, confirm prior physics predictions, and detail how supersymmetry (g=2) yields γ=2πβ across scales, while highlighting regimes where the hard-disk limit or other interactions modify the effective coupling. This work thus provides a solid mathematical foundation for the emergent CSS description of almost-bosonic anyons and clarifies the role of spin-orbit coupling and flux regularization in the derivation.

Abstract

In this work we consider the $N$-body Hamiltonian describing the microscopic structure of a quantum gas of almost-bosonic anyons. This description includes both extended magnetic flux and spin-orbit/soft-disk interaction between the particles which are confined in a scalar trapping potential. We study a physically well-motivated ansatz for a sequence of trial states, consisting of Jastrow repulsive short-range correlations and a condensate, with sufficient variational freedom to approximate the ground state (and possibly also low-energy excited states) of the gas. In the limit $N \to \infty$, while taking the relative size of the anyons to zero and the total magnetic flux $2πβ$ to remain finite, we rigorously derive the stationary Chern-Simons-Schrödinger/average-field-Pauli effective energy density functional for the condensate wave function. This includes a scalar self-interaction parameter $γ$ which depends both on $β$, the diluteness of the gas, and the spin-orbit coupling strength $g$, but becomes independent of these microscopic details for a particular value of the coupling $g=2$ in which supersymmetry is exhibited (on all scales, both microscopic and mesoscopic) with $γ=2π|β|$. Our findings confirm and clarify the predictions we have found in the physics literature.

Microscopic derivation of the stationary Chern-Simons-Schrödinger equation for almost-bosonic anyons

TL;DR

The paper rigorously derives a stationary Chern-Simons-Schrödinger (CSS) energy functional as the effective description of a two-dimensional quantum gas of almost-bosonic anyons with flux attachment and spin-orbit coupling. By introducing a Hartree-Jastrow trial state and a precise scaling limit where the anyon radius R_N→0 and the anyon-flux parameter α_N scales as α_N=β/(N−1), the authors show that, for β_N=(N−1)α_N→β>0 and ω_N→ω∈[0,∞], the energy per particle converges to the CSS energy density ∫|(-i∇+βA[|u|^2])u|^2 + ∫V|u|^2 + 2πβ G(2βω,g)∫|u|^4, with the one-particle density tending to |u|^2. The key technical achievement is a rigorous construction of the two-body Jastrow factor f and meticulous control of kinetic, potential, singular, and current terms to obtain the exact leading-order contribution, including the scale-dependent function G that encodes the two-body physics and the spin-orbit coupling g. The results connect microscopic parameters to macroscopic, self-consistent magnetic interactions, confirm prior physics predictions, and detail how supersymmetry (g=2) yields γ=2πβ across scales, while highlighting regimes where the hard-disk limit or other interactions modify the effective coupling. This work thus provides a solid mathematical foundation for the emergent CSS description of almost-bosonic anyons and clarifies the role of spin-orbit coupling and flux regularization in the derivation.

Abstract

In this work we consider the -body Hamiltonian describing the microscopic structure of a quantum gas of almost-bosonic anyons. This description includes both extended magnetic flux and spin-orbit/soft-disk interaction between the particles which are confined in a scalar trapping potential. We study a physically well-motivated ansatz for a sequence of trial states, consisting of Jastrow repulsive short-range correlations and a condensate, with sufficient variational freedom to approximate the ground state (and possibly also low-energy excited states) of the gas. In the limit , while taking the relative size of the anyons to zero and the total magnetic flux to remain finite, we rigorously derive the stationary Chern-Simons-Schrödinger/average-field-Pauli effective energy density functional for the condensate wave function. This includes a scalar self-interaction parameter which depends both on , the diluteness of the gas, and the spin-orbit coupling strength , but becomes independent of these microscopic details for a particular value of the coupling in which supersymmetry is exhibited (on all scales, both microscopic and mesoscopic) with . Our findings confirm and clarify the predictions we have found in the physics literature.

Paper Structure

This paper contains 30 sections, 21 theorems, 253 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Mathematical main result
  4. Discussion: origins and consequences for physics
  5. Pointlike and ideal anyons
  6. Extended anyons and average-field approximation
  7. Spin-orbit coupled Hamiltonian
  8. The almost-bosonic average-field approximation
  9. The Chern--Simons--Schrödinger energy
  10. Discussion: results
  11. The two-body problem
  12. The motivation for the Jastrow pair correlation
  13. The case $R=0$
  14. The case $R>0$ and $g \neq 0$
  15. A simplified repulsive ansatz for $R>0$
  16. ...and 15 more sections

Key Result

Theorem 1.1

For each integer $N \ge 2$, let the Hamiltonian $H_N$ be defined as in def:HN with the $N$-dependent parameters $\alpha=\alpha_N \ge 0$, $R=R_N \ge 0$, and $g=g_N \geq 0$. Consider $\Psi_N^{\mathrm{trial}}$ as the trial state given in def:trial with $b_N>R_N$ and $u\in C^{\infty}_c(B(0,R_1))$ for so and assume that Then, together with

Figures (1)

  • Figure 1: Sketch of $-\gamma_*(\beta)$ with exact $\mathrm{NLL}$s at $\beta \in 2\mathbb{Z} \setminus \{0\}$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2: Sub-critical/polynomial limit
  • Remark 1.3: Scattering length
  • Remark 1.4: Supersymmetry and scale invariance
  • Remark 1.5: Hard-disk limit
  • Remark 1.6: Simplified Jastrow ansatz
  • Remark 1.7: Attractive case and stability
  • Remark 1.8
  • Proposition 2.1: Two-body scattering energy
  • proof
  • ...and 36 more