Bounding the excitation gap of the Laughlin and other model incompressible fractional quantum Hall (FQH) states.

The Laughlin and other model FQH states have the property that (a) they are exact eigenstates (and are highest-density kernel or zero-energy eigenstates) of model "pseudopotential" Hamiltonians, and (b) are related to (chiral) conformal blocks of (Euclidean) 2D conformal field theories (cft). These model Hamiltonians have a non-trivial kernel spanned by a basis of quasi-hole states related to a chiral cft. What is still lacking is information about the non-kernel (quasiparticle) eigenstates of these models, and the persistence of the energy gap in the thermodynamic limit.They have analogies to the AKLT spin-chain model, for which such questions have been answered. The question of proving (or disproving) a lower bound to the spectral gap separating kernel and non-kernel states becomes more interesting because certain of these models (e.g., the "Gaffnian") are related to a non-unitary cft which is now believed to mean that they have gapless bulk excitations in the thermodynamic limit, and perhaps represent critical states at continuous transitions between topologically-distinct FQH states. I will discuss these fundamental questions, and review strategies and progress towards answering them.