Spin of fractional quantum Hall neutral modes and "missing states" on a sphere

TL;DR

The paper investigates why certain low-angular-momentum states are absent in the sphere geometry for neutral excitations in fractional quantum Hall systems. By treating a spin-$s$ neutral quasiparticle as acquiring a Berry phase from the spin connection on a sphere, the authors map the problem to a fictitious magnetic monopole with charge $g=s/e$, leading to energy levels $E_L = [L(L+1)-(eg)^2]/(2mR^2)$ and a constraint $L\ge |s|$. This geometric perspective explains the observed missing $L=0$ (and $L=1$ for $s=2$) in the long-wavelength magnetoroton and extends to other neutral modes, such as neutral fermions with $s=3/2$ and interlayer spin-1 excitations, while certain Goldstone modes do not exhibit missing states. The framework provides a coherent interpretation of sphere-spectrum data, informs counting of neutral modes in Jain sequences, and connects to broader geometric response ideas in FQH physics, with potential relevance to experiments and numeric studies in related systems like FCIs.

Abstract

A low-energy neutral quasiparticle in a fractional quantum Hall system appears in the latter's energy spectrum on a sphere as a series of many-body excited states labeled by the angular momentum $L$ and whose energy is a smooth function of $L$ in the limit of large sphere radius. We argue that the signature of a nonvanishing spin (intrinsic angular momentum) $s$ of the quasiparticle is the absence, in this series, of states with total angular momentum less than $s$.We reinterpret the missing of certain states, observed in an exact-diagonalization calculation of the spectrum of the $ν=7/3$ FQH state in a wide quantum well as well as in many proposed wave functions for the excited states as a consequence of the spin-2 nature of the zero-momentum magnetoroton.