Geometry induced quantum Hall effect and Hall viscosity.

For a particle confined to the 2D helical surface embedded in 4D Euclidean space, we deduce the effective Hamiltonian in the thin-layer quantization formalism. We find that the gauge structure of the effective dynamics is determined by the torsion that plays the role of U(1) gauge potential, and the topological structure of associated states is defined by the orbital spin that originates from the 4D space. Strikingly, the response to torsion contributes quantum Hall effect, and the response to the deformation of torsion provides Hall viscosity that is perfectly presented as simultaneous occurrence of multiple channels for quantum Hall effect. The result can directly provide a way to probe the Hall viscosity.