Noncommutative geometry for three-dimensional topological insulators

We generalize the noncommutative relations obeyed by the guiding centers in the two-dimensional quantum Hall effect to those obeyed by the projected position operators in three-dimensional (3D) topological band insulators. The noncommutativity in 3D space is tied to the integral over the 3D Brillouin zone of a Chern-Simons invariant in momentum-space. We provide an example of a model on the cubic lattice for which the chiral symmetry guarantees a macroscopic number of zero-energy modes that form a perfectly flat band. This lattice model realizes a chiral 3D noncommutative geometry. Finally, we find conditions on the density-density structure factors that lead to a gapped 3D fractional chiral topological insulator within Feynman's single-mode approximation.