Anisotropic quantum Hall droplets challenge the plasma analogy

I will present recent work on integer quantum Hall droplets with anisotropic trapping potentials. Using semiclassical methods, we obtain the one-particle energy spectrum and wave functions in the lowest Landau level by deriving and solving a transport equation inspired by standard WKB theory. This shows that energy eigenstates are localized on equipotentials of the trap, generalizing the rotational-symmetric case for isotropic potentials. From these microscopic first-principle considerations, we show that many-body correlations along the droplet's edge are long-ranged, in agreement with low-energy edge modes described by a free chiral conformal field theory in terms of the canonical angle variable of the potential. Comparing our results with the widely used plasma analogy, we show that the latter is unreliable at predicting edge properties of quantum Hall states. This discrepancy arises from a difference in geometries between quantum Hall droplets and plasmas (Coulomb gases): The former are incompressible liquids subject to area-preserving deformations, while the latter are governed by electrostatics and thus involve conformal maps. Consequently, the plasma analogy generally fails at the edge of anisotropic droplets.