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

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 are recovered.