Ground state wave functions for the quantum Hall effect on a sphere and the Atiyah–Singer index theorem

Dolan, Brian (2020) Ground state wave functions for the quantum Hall effect on a sphere and the Atiyah–Singer index theorem. Journal of Physics A: Mathematical and Theoretical, 53. p. 215306. ISSN 1751-8113 Online ISSN 1751-8121 (Preprint)

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Official URL: http://iopscience.iop.org/1751-8121

Abstract

The quantum Hall effect is studied in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field, which is supplied by a Wu–Yang magnetic monopole at the center of the sphere. Wave functions are cross-section of a non-trivial U(1) bundle, the zero point energy then vanishes and no perturbations can lower the energy. The Atiyah–Singer index theorem constrains the degeneracy of the ground state. The fractional quantum Hall effect is also studied in the composite Fermion model. Vortices of the statistical gauge field are supplied by Dirac strings associated with the monopole field. A unique ground state is attained only if the vortices have an even number of flux units and act to counteract the background field, reducing the effective field seen by the composite fermions. There is a unique gapped ground state and, for large particle numbers, fractions $\nu =\frac{1}{2k+1}$ are recovered.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 08 Apr 2022 13:36
Last Modified: 15 Dec 2022 09:05
URI: https://dair.dias.ie/id/eprint/1183

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