## Shift class

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In a Quantum Hall fluid on the plane the number of electrons $N_e$ and the number of magnetic flux quanta $N_{\phi }$ are simply related by $\nu N_{\phi }=N_e$ (where $\nu$ is by definition the filling factor). On a closed topological 2-manifold, such as a sphere, there is a shift $S$ in the relation between $N_e$ and $N_{\phi }$
It was conjectured that shift is related to new quantum numbers (spin vector) for the Hall fluid, representing orbital spin degrees of freedom. Spin vectors are quantized. In the absence of impurities, two Hall fluids with different spin vectors cannot change into each other without a phase transition and closing of the energy gap. In principle the spin vector can be measured through its coupling to the curvature of space.
For original reference see X. G. Wen, A. Zee Phys. Rev. Lett. 69, 953 - 956 (1992) Shift and spin vector: New topological quantum numbers for the Hall fluids