Nonlinear Landau levels in the almost-bosonic anyon gas

TL;DR

The paper studies an in-plane gas of abelian anyons in a trapping potential in the almost-bosonic limit where the total flux $\beta$ is of order 1, and derives an effective Chern-Simons–Schrödinger (CSS) energy functional $\mathcal{E}_{\beta,\gamma,V}[\phi]$ with self-generated magnetic field $B_\rho = \mathrm{curl}\,\mathbf{A}_\rho = 2\pi\rho$. A Hartree–Jastrow ansatz reduces the many-body problem to this mean-field description, yielding a two-parameter theory with $\beta$ (total flux units) and $\gamma$ (effective spin-orbit self-interaction) and establishing stability criteria; nonlinear Landau levels (NLLs) appear at $\gamma = -2\pi|\beta|$, corresponding to zero-energy Jackiw–Pi self-dual solitons and a supersymmetry-breaking structure. The work derives precise relations among physical parameters, including $\gamma = 2\pi \beta \frac{1+\frac{g}{2}-(1-\frac{g}{2})e^{-2\beta\sigma}}{1+\frac{g}{2}+(1-\frac{g}{2})e^{-2\beta\sigma}}$, and demonstrates that the spectrum in the large-$\beta$ regime features locally homogeneous vortex lattices with a coarse-grained Thomas–Fermi–type density. Numerically and analytically, the authors connect the CSS spectrum to exact NLL solutions and illustrate a transition toward enhanced stability via counter-rotating vortices, while clarifying relations between various anyon models and highlighting a novel supersymmetry-breaking phenomenon with potential implications for many-anyon spectral problems and intermediate quantum statistics.

Abstract

We consider the quantitative description of a many-particle gas of interacting abelian anyons in the plane, confined in a trapping potential. If the anyons are modeled as bosons with a magnetic flux attachment, and if the total magnetic flux is small compared to the number of particles, then an average-field description becomes appropriate for the low-energy collective state of the gas. Namely, by means of a Hartree-Jastrow ansatz, we derive a two-parameter Chern-Simons-Schrödinger energy functional which extends the well-known Gross-Pitaevskii / nonlinear Schrödinger density functional theory to the magnetic (anyonic) self-interaction. One parameter determines the total number of self-generated magnetic flux units in the system, and the other the effective strength of spin-orbit self-interaction. This latter interaction can be either attractive/focusing or repulsive/defocusing, and depends both on the intrinsic spin-orbit interaction and the relative length scale of the flux profile of the anyons. Densities and energies of ground and excited states are studied analytically and numerically for a wide range of the parameters and align well with a sequence of exact nonlinear Landau levels describing Jackiw-Pi self-dual solitons. With increasing flux, counter-rotating vortices are formed, enhancing the stability of the gas against collapse. Apart from clarifying the relations between various different anyon models that have appeared in the literature, our analysis sheds considerable new light on the many-anyon spectral problem, and also exemplifies a novel supersymmetry-breaking phenomenon.

Figures (4)

• Figure 1: Density and phase of an approximate minimizer for $E(\beta=10,\gamma=20\pi,V_\mathrm{h}) \approx 9.066$.
• Figure 2: Stability holds if and only if $E(\beta,\gamma,V) > -\infty$. NLLs exist at $\beta \in 2\mathbb{Z} \setminus \{0\}$ with the critical coupling $\gamma = -2\pi|\beta|$.
• Figure 3: Analytical and numerical energies for attractive gas $\gamma = 2\pi\beta\kappa < 0$ at $\kappa = -1$ ("kreinyons") resp. $\kappa = -0.9$.
• Figure 4: Analytical and numerical energies for repulsive gas $\gamma = 2\pi\beta\kappa \ge 0$ at $\kappa=0$ resp. 1 (ideal anyons). See the text.