Abstract
We study two-dimensional (2D) droplets of noninteracting electrons in a strong magnetic field, placed in a confining potential with arbitrary shape. Using semiclassical methods adapted to the lowest Landau level, we obtain near-Gaussian energy eigenstates that are localized on level curves of the potential and have a position-dependent height. This one-particle insight allows us to deduce explicit formulas for expectation values of local many-body observables, such as density and current, in the thermodynamic limit. In particular, correlations along the edge are long-ranged and inhomogeneous. As we show, this is consistent with the system’s universal low-energy description as a free 1D chiral conformal field theory of edge modes, known from earlier works in simple geometries. A delicate interplay between radial and angular dependencies of eigenfunctions ultimately ensures that the theory is homogeneous in terms of the canonical angle variable of the potential, despite its apparent inhomogeneity in terms of more naïve angular coordinates. Finally, we propose a scheme to measure the anisotropy by subjecting the droplet to microwave radiation; we compute the corresponding absorption rate and show that it depends on the droplet’s shape and the waves’ polarization. These results, both local and global, are likely to be observable in solid-state systems or quantum simulators of 2D electron gases with a high degree of control on the confining potential.
- Received 31 March 2023
- Revised 22 September 2023
- Accepted 6 November 2023
DOI:https://doi.org/10.1103/PhysRevX.14.011030
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Quantum Hall droplets are 2D electron fluids in strong magnetic fields. They provide the paradigmatic example of topological phases of matter, with an emergent invariance under geometric deformations. However, the study of deformed droplets is often limited to nearly isotropic cases or poorly controlled low-energy approximations, despite ongoing experimental efforts to create droplets of arbitrary shapes. Here, we present a first-principles, microscopic study of anisotropic quantum Hall droplets, well away from the isotropic situations investigated so far.
We achieve this by developing a general semiclassical approach for quantum Hall wave functions in arbitrary, anisotropic confining potentials. This allows us to compute many-body observables in anisotropic droplets, yielding remarkably explicit predictions, for example, for electronic density and correlations. In particular, we show that low-energy excitations are described by a homogeneous gapless field theory on the droplet’s edge, despite the fact that the underlying microscopic wave functions are highly inhomogeneous.
Our predictions bridge theory and experiments, as we expect that they will be verifiable in several ways. One is direct imaging close to the edge of quantum Hall droplets. Another is microwave spectroscopy, since our study predicts the appearance of a series of distinctive peaks of absorption whose magnitude is directly linked to the droplet’s geometry. Furthermore, our semiclassical formalism paves the way for new research avenues on nonlinear edge modes and anisotropic fractional quantum Hall states, whose analytical study has been challenging until now.
