Density algebra for three-dimensional topological insulators

Details Density algebra for three-dimensional topological insulators Density algebra for three-dimensional topological insulators

2012-02-23

arxiv.org/abs/1202.5188v1

Physics

Condensed Matter

Strongly Correlated Electrons

28 pages, 3 figures

Scientific paper

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.

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