Magnetic Thomas-Fermi theory for 2D abelian anyons

TL;DR

This work develops a mean-field description for two-dimensional abelian anyons in a trapping potential by coupling fermions to a self-generated magnetic field, leading to a Chern-Simons-Schrödinger–type system. It introduces a Hartree functional with a density-dependent vector potential $\mathbf{A}[\rho]$ and analyzes its Euler–Lagrange equations, along with magnetic Thomas-Fermi (mTF) and local density approximations to capture large-$N$ behavior. The authors demonstrate that, in the almost-fermionic limit and in the high-density mTF regime, the theory yields correct qualitative and quantitative trends for energies and densities, with position densities largely α-independent and momentum densities revealing signatures of anyon statistics. Numerical simulations corroborate the analytical predictions, showing good agreement with TF-type theories and highlighting momentum-space observables as promising experimental signatures for anyonic statistics in cold-atom realizations.

Abstract

Two-dimensional abelian anyons are, in the magnetic gauge picture, represented as fermions coupled to magnetic flux tubes. For the ground state of such a system in a trapping potential, we theoretically and numerically investigate a Hartree approximate model, obtained by restricting trial states to Slater determinants and introducing a self-consistent magnetic field, locally proportional to matter density. This leads to a fermionic variant of the Chern-Simons-Schr{ö}dinger system. We find that for dense systems, a semi-classical approximation yields qualitatively good results. Namely, we derive a density functional theory of magnetic Thomas-Fermi type, which correctly captures the trends of our numerical results. In particular, we explore the subtle dependence of the ground state with respect to the fraction of magnetic flux units attached to particles.

Figures (9)

• Figure 1: Plot $\alpha \mapsto c(\alpha) - 1$ of the anyonic magnetic Thomas-Fermi constant \ref{['eq:TF constant']}, relative to the free fermion case, i.e. fermions at $\alpha=0$ and bosons at $\alpha=1$. Also shown (dashed) is the upper bound $\alpha^2/4$.
• Figure 2: Position density (normalized), $N=100$, $\alpha=\frac{1.5}{\sqrt{N}}=0.15$, $V(\mathbf{x})=|\mathbf{x}|^{2}$.
• Figure 3: Momentum density (rescaled), $N=100$, $\alpha=\frac{1.5}{\sqrt{N}}=0.15$, $V(\mathbf{x})=|\mathbf{x}|^{2}$.
• Figure 4: Position density (normalized), $N=100$, $\alpha=\frac{1.5}{\sqrt{N}}=0.15$, $V(\mathbf{x})=|\mathbf{x}|^{4}$.
• Figure 5: Momentum density (rescaled), $N=100$, $\alpha=\frac{1.5}{\sqrt{N}}=0.15$, $V(\mathbf{x})=|\mathbf{x}|^{4}$.