1202.5188 (Titus Neupert et al.)

Density algebra for three-dimensional topological insulators    [PDF]

Titus Neupert, Luiz Santos, Shinsei Ryu, Claudio Chamon, Christopher Mudry

We present the algebra for density operators projected to a topological band
of a three-dimensional (3D) system. This algebra generalizes to 3D space the
Girvin-MacDonald-Platzman algebra for the densities projected to the lowest
Landau level in the case of the 2D fractional quantum Hall effect. We provide
an example of a model on the cubic lattice in which the chiral symmetry
guarantees a macroscopic number of zero-energy modes that form a perfectly flat
band, and explicitly construct the algebra for the density operators projected
onto this topological dispersionless band. The algebra of the projected density
operators is related to the emergence of noncommutativity of the spatial
coordinates of particles propagating in 3D, similarly to the noncommutativity
of coordinates projected to the lowest Landau level in 2D. The noncommutativity
in 3D is tied to a nonvanishing theta-term associated to the integral over the
3D Brillouin zone of a Chern-Simons invariant in momentum-space. 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.

View original: http://arxiv.org/abs/1202.5188