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arXiv · 1202.5188

Noncommutative geometry for three-dimensional topological insulators

Titus Neupert·Luiz Santos·Shinsei Ryu·Claudio Chamon·Christopher Mudry
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Abstract

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.

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Titus Neupert, Luiz Santos, Shinsei Ryu, Claudio Chamon, Christopher Mudry. 2012-02-23. Noncommutative geometry for three-dimensional topological insulators. https://doi.org/10.1103/physrevb.86.035125

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