In this talk, we discuss the bulk gap for a truncated 1/3-filled Haldane pseudopotential for the fractional quantum Hall effect. For this Hamiltonian with periodic boundary conditions, we establish a spectral gap above the highly degenerate ground-state space which is uniform in the volume and particle number. These bounds are proved by identifying invariant subspaces to which we apply gap-estimate methods previously developed only for quantum spin Hamiltonians. In the case of open boundary conditions, a lower bound on the spectral gap accurately reflects the presence of edge states, which do not persist into the bulk. Customizing the gap technique to the invariant subspace, we avoid the edge states and establish a more precise estimate on the bulk gap in the case of periodic boundary conditions. The same approach can also be applied to prove a bulk gap for the analogously truncated Haldane pseudopotential with maximal half filling, which describes a strongly correlated system of spinless bosons in a cylinder geometry.
This is based off joint work with S. Warzel.
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