Haldane FQHS gap conjecture for Haldane pseudopotentials

Establish a uniform positive lower bound, independent of system size, for the excitation gap above the ground-state energy (the neutral spectral gap) of the two-body Haldane pseudopotential Hamiltonians at filling fraction 1/q for small q ∈ {2, 3, …} in fractional quantum Hall systems.

Background

The paper studies translation-invariant, charge- and dipole-conserving many-body Hamiltonians at fractional filling and proves that the charge gap dominates the neutral gap. Within this framework, a central outstanding problem in fractional quantum Hall physics concerns the spectral gap of parent Hamiltonians for Laughlin states.

The two-body Haldane pseudopotential Hamiltonians are canonical parent Hamiltonians for 1/q Laughlin states. It is widely believed that for small q the excitation gap above the ground state (the neutral gap) remains bounded below by a strictly positive constant uniformly in system size—this is Haldane’s FQHS gap conjecture. The authors note that, although this conjecture has been proven for truncated pseudopotentials, it remains unresolved for the full Haldane pseudopotential models. They also outline how their gap comparison method could provide a concrete route to attack this conjecture.