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arXiv:2504.13481 · SciPost Physics Core

Magnetic Thomas-Fermi theory for 2D abelian anyons

Antoine Levitt, Douglas Lundholm, Nicolas Rougerie

Published Mar 30, 2026• 1 citation
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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-Schrödinger 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.

DOI
10.21468/scipostphyscore.9.1.018

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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-Schrödinger 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.
We develop a density functional treatment of noninteracting Abelian anyons, which is capable, in principle, of dealing with a system of a large number of anyons in an external potential. Comparison with exact results for a few particles shows that the model captures the behavior qualitatively and semiquantitatively, especially in the vicinity of the fermionic statistics. We then study anyons with statistics parameter 1+1/n1+1/n1+1/n, which are thought to condense into a superconducting state. An indication of the superconducting behavior is the mean-field result that, for uniform density systems, the ground-state energy increases under the application of an external magnetic field independent of its direction. Our density functional theory based analysis does not find that to be the case for finite systems of anyons, which can accommodate a weak external magnetic field through density transfer between the bulk and the boundary rather than through transitions across effective Landau levels, but the ``Meissner repulsion'' of the external magnetic field is recovered in the thermodynamic limit as the effect of the boundary becomes negligible. We also consider the quantum Hall effect of anyons, and show that its topological properties, such as the charge and statistics of the excitations and the quantized Hall conductance, arise in a self-consistent fashion.
This thesis addresses the dimensional reduction of two-dimensional (2D) Abelian anyons under strong anisotropic confinement, studied both at the microscopic many-body level and at the mean-field/almost-bosonic level.Anyons are quasi-particles with fractional statistics that interpolate between bosons and fermions, characterized by an exchange phase e^{i·π·α} for α in (0, 1). The theoretical concept of anyons dates back to the work of Leinaas and Myrheim in 1977, while convincing experimental evidence for their existence emerged only in 2020, through their realization in fractional quantum Hall systems. Investigating their behavior under confinement not only clarifies the connections between one- and two-dimensional anyons, but also offers new insights into the mathematical structure of anyonic systems.At the microscopic level, we consider N anyons in the magnetic gauge picture with a harmonic trapping potential that freezes the motion in the longitudinal (y) direction. We rigorously prove that, after subtracting the frozen longitudinal contribution, the associated 2D energy Γ-converges to that of the one-dimensional (1D) Tonks-Girardeau (TG) gas. This establishes a precise mathematical connection between 2D anyons and 1D impenetrable bosons, and, via the Bose-Fermi mapping, to a free Fermi gas. Importantly, this 1D limit model is independent of the exchange parameter α at leading order, indicating that the effect of the magnetic gauge field is gauged away up to o(1) corrections. Equivalently, the long-range Aharonov-Bohm interactions collapse into a hard-core exclusion constraint in this dimensional reduction limit.At the mean-field level, in the average-field regime (N → ∞ with α proportional to N^{-1}), the 2D anyon system is described by the Chern-Simons-Schrödinger (CSS) functional, a nonlinear magnetic Schrödinger model with a self-consistent gauge field. Performing the same anisotropic dimensional reduction limit, the resulting 1D effective model is described by a quintic nonlinear Schrödinger (NLS) equation, whose quintic nonlinearity stems from the magnetic self-consistent gauge in the CSS model. This dimensional reduction is established both at the level of ground-state energies and, under suitable well-posedness assumptions on the CSS dynamics, at the level of time evolution. Unlike the TG gas, the effective coupling constant g in the quintic NLS retains a dependence on the exchange parameter α through g = π·β² for β proportional to α·N.Taken together, these results yield two distinct effective 1D descriptions arising from different orders of limits. Crucially, the mean-field/almost-bosonic limit and the dimensional reduction (strong confinement) limit do not commute: the TG gas limit is α-independent, whereas the quintic NLS limit keeps an α-dependence through β proportional to α·N. This non-commutativity highlights the subtle interplay between scaling limits and confinement in anyonic systems. Future work might investigate whether a critical regime separates these two situations.
The energy density is computed for a U(2) Chern-Simons theory coupled to a non-relativistic fermion field (a theory of ``non-Abelian anyons'') under the assumptions of uniform charge and matter density. When the matter field is a spinless fermion, we find that this energy is independent of the two Chern-Simons coupling constants and is minimized when the non-Abelian charge density is zero. This suggests that there is no spontaneous breaking of the SU(2) subgroup of the symmetry, at least in this mean-field approximation. For spin-1/2 fermions, we find self-consistent mean-field states with a small non-Abelian charge density, which vanishes as the theory of free fermions is approached.