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

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 $ν=1/3$ 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.