We introduce an elementary inequality relating the charge and neutral gaps of a family of quantum many-body systems and discuss an application to the so-called pseudo-potentials modeling fractional quantum Hall systems. These are quantum many-body Hamiltonians that are frustration free and have two symmetries, one related to the conservation of charge (particle number) and another to the conservation of dipole moment (angular momentum), in addition to translation invariance. We show that for such systems the general inequality can be refined, opening the possibility of new applications to the study of the spectral gap of such systems.
(Joint work with Marius Lemm, Simone Warzel, and Amanda Young, arxiv:2410.11645).