A. Zee, Quantum Field Theory in a Nutshell · Problem VI.2.1

Field Theory Redux

Physics — QFT

Problem

To define filling factor precisely, we have to discuss the quantum Hall system on a sphere rather than a plane. Put a magnetic monopole of strength GG (which according to Dirac can be only a half-integer or an integer) at the center of a unit sphere. The flux through the sphere is equal to Nϕ=2GN_\phi = 2G. Show that the single electron energy is given by El=(12ωc)l(l+1)−G2/GE_l = (\frac{1}{2} \omega_c) \sqrt{l(l + 1) - G^2}/G with the Landau levels corresponding to l=G,G+1,G+2,…l = G, G + 1, G + 2, \ldots, and that the degeneracy of the llth level is 2l+12l + 1. With LL Landau levels filled with noninteracting electrons (ν=L\nu = L) show that Nϕ=ν−1Ne−SN_\phi = \nu^{-1} N_e - S, where the topological quantity SS is known as the shift.
Intuitive explanation

The quantum Hall effect on a sphere reveals deep topological structure hidden on a plane. A magnetic monopole at the center quantizes orbital angular momentum ll rather than Landau level index, with the sphere's geometry enforcing specific allowed values and degeneracies. The key insight is that the shift SS emerges from the mismatch between the total flux NϕN_\phi and electron count NeN_e—it's a topological invariant that counts how many flux quanta are "used up" by the geometry and quantum correlations rather than by electrons alone.

Intuitive explanation

The quantum Hall effect on a sphere reveals deep topological structure hidden on a plane. A magnetic monopole at the center quantizes orbital angular momentum ll rather than Landau level index, with the sphere's geometry enforcing specific allowed values and degeneracies. The key insight is that the shift SS emerges from the mismatch between the total flux NϕN_\phi and electron count NeN_e—it's a topological invariant that counts how many flux quanta are "used up" by the geometry and quantum correlations rather than by electrons alone.

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