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Local Commuting Projector Hamiltonians and the Quantum Hall Effect

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Abstract

We prove that neither Integer nor Fractional Quantum Hall Effects with nonzero Hall conductivity are possible in gapped systems described by Local Commuting Projector Hamiltonians.

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Notes

  1. In many papers “holonomies” are called fluxes. We find this terminology confusing, since the word “flux" is also used to describe a region of nonzero magnetic field, while \(\beta _x,\beta _y\) parameterize a flat U(1) gauge field on a torus.

  2. It is usually assumed that the ground-states have \(N=0\), and thus all \(n_\Gamma \) vanish. We allow for more general possibilities.

  3. Strictly speaking, Cartier divisors, but on smooth algebraic varieties this is the same as Weil divisors [12], so we do not need to make a distinction.

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Correspondence to Lukasz Fidkowski.

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Communicated by N. Nekrasov.

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Kapustin, A., Fidkowski, L. Local Commuting Projector Hamiltonians and the Quantum Hall Effect. Commun. Math. Phys. 373, 763–769 (2020). https://doi.org/10.1007/s00220-019-03444-1

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  • DOI: https://doi.org/10.1007/s00220-019-03444-1

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