Torus geometry eigenfunctions of an interacting multi-Landau level Hamiltonian

TL;DR

This paper extends a multi-Landau-level model Hamiltonian to torus and cylinder geometries, providing exact spectra and eigenfunctions for strongly interacting electrons in quantum Hall systems, revealing topological degeneracies and excitation structures.

Contribution

It generalizes the exact eigenfunctions and spectra of a multi-Landau-level Hamiltonian from disk to torus and cylinder geometries, enabling analysis of quantum Hall excitations in these settings.

Findings

Exact spectra on torus match integer quantum Hall spectra up to topological degeneracy.

Eigenfunctions for charged excitations are constructed on torus and sphere geometries.

Extension of eigenfunctions to cylinder geometry is demonstrated.

Abstract

A short-ranged, rotationally symmetric multi-Landau-level model Hamiltonian for strongly interacting electrons in a magnetic field was proposed [A. Anand et al, Phys. Rev. Lett. 126, 136601 (2021)] with the key feature that it allows exact many-body eigenfunctions on the disk not just for quasiholes but for all charged and neutral excitations of the entire Jain sequence filling fractions. We extend this to geometries without full rotational symmetry, namely the torus and cylinder geometries, and present their spectra. Exact diagonalization of the interaction on the torus produces the low-energy spectra at filling fraction $\nu=n/(2pn+1)$ that is identical, up to a topological $(2pn+1)$-fold multiplicity, to that of the integer quantum Hall spectra at $\nu=n$, for the incompressible state as well as all excitations. While the ansatz eigenfunctions in the disk geometry cannot be generalized to closed geometries such as torus or sphere, we show how to extend them to cylinder geometry. Meanwhile, we show eigenfunctions for charged excitations at filling fractions between $\frac{1}{3}$ and $\frac{2}{5}$ can be written on the torus and the spherical geometries.