IQUIST Seminar: Amanda Young, University of Illinois Urbana-Champaign

Haldane pseudopotentials were first proposed as Hamiltonian models for the fractional quantum Hall effect, and it has been long expected that they should exhibit the characteristic properties of this exotic phase of matter, including a bulk spectral gap above the ground state energy. While this conjecture remains open, it was recently verified for a truncated version of the 1/3-filled Haldane pseudopotential in the cylinder geometry. In this case, numerical evidence suggests the presence of edge models that close the gap of the truncated model with open boundary conditions closed as the cylinder radius tends to zero; in contrast, for periodic boundary conditions, the gap remains open in the same radius limit. One difficulty in verifying these results analytically is that the standard scheme for establishing a non-zero spectral gap for a model with periodic boundary conditions naturally produces a lower bound on the bulk gap that still reflects the energy of the edge modes. To obtain an estimate on the bulk gap that reflects its true behavior, a new gap estimating strategy based on invariant subspaces was developed. By customizing the spectral gap method to key invariant subspaces of the Hamiltonian, one is able to successfully avoid the edge states and produce a more accurate lower bound on the bulk gap. In this talk, we discuss this invariant subspace strategy for proving bulk gaps in the presence of edge states. This is based off joint work with S. Warzel.

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