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.

Figure 1

Assumed spectral gap in the single-particle energy spectrum. Here, ${\mu }_{\mathrm{c}}$ denotes the chemical potential and the insulating noninteracting many-body ground state $|\Phi \rangle$ is obtained by filling all the states in the $\tilde{N}$ bands below the spectral gap.

Figure 2

Energy spectrum of the lattice model defined in Eq. (3.21) for different values of the parameter $M$. Panels (b) and (d) show the gap-closing topological transitions. Note the dispersionless band at zero energy in each spectrum. The spectrum is plotted along the straight path connecting the following points in the BZ: $\Gamma =(0,0,0)$, $X=(0,\pi ,0)$, $M=(\pi ,\pi ,0)$, and $R=(\pi ,\pi ,\pi )$.

Figure 3

Numerical evaluation of the topological invariant ${\theta }^{\left(0\right)}=\pi \nu \left(M\right)$ (solid line) for the model (3a) The parameters ${\theta }^{\prime }$ and ${\theta }^{\prime \prime }$ that sum up to the topological invariant $\theta$ are defined in Eqs. (C8a) and(C8d), respectively.