Excitation and tunneling spectra of a fractional quantum Hall system in the thin cylinder limit

TL;DR

This paper develops an analytical method using perturbation theory in the thin cylinder limit to study excitations in the fractional quantum Hall effect, linking theoretical predictions with recent STM experimental results.

Contribution

It introduces a systematic approach to enumerate low-lying excitations in the FQHE at $ u=1/3$ using the thin cylinder limit and connects this to STM spectra analysis.

Findings

Predicts significant dispersion of neutral excitations in the thin cylinder limit.

Shows charged excitations produce sharp LDOS spectra, consistent with experiments.

Numerical results suggest charged excitations are confined to narrow energy ranges.

Abstract

The excitations of fractional quantum Hall effect (FQHE) states have been largely inaccessible to experimental probes until recently. New electron scanning tunneling microscopy (STM) results from Hu et.al. (arXiv:2308.05789) show promise in detecting and identifying these excited states via the local density of states (LDOS) spectrum. On a torus, there exists a mapping {from the lowest Landau level states} to a 1D lattice {with a Hamiltonian that features} dipole moment conservation. In this work, we apply perturbation theory starting from the thin cylinder limit ($L_x \rightarrow \infty, L_y <l_B$ for torus dimensions $L_x$ and $L_y$ {and magnetic length $l_B$}) to obtain an analytical approach to the low-lying neutral and charged excitations of the $\nu =1/3$ FQHE state. Notably, in the thin cylinder we can systematically enumerate all the low-lying excitations by the patterns of 'dipoles' formed by the electron occupation pattern on the 1D lattice. We find that the thin-cylinder limit predicts a significant dispersion of the low-lying neutral excitations but sharpness of the LDOS spectra, which measure charged excitations. We also discuss connections between our work and several different approaches to the FQHE STM spectra, including those using the composite fermion theory. Numerical exact diagonalization beyond the thin-cylinder limit suggests that the energies of charged excitations remain largely confined to a narrow range of energies, which in experiments might appear as a single peak.