| Electronic density and position operators, when projected to a subspace of the single-particle Hilbert space, for example an isolated energy band, obey in general a non-commutative algebra. As noted by Girvin, McDonald and Platzman, such an algebra plays a crucial role in Landau levels, where it is responsible for the properties of both integer and fractional quantum Hall states. In my talk, I will present a generalization of this non-commutative algebra approach to three-dimensional systems and in particular relate it to the topological attributes of non-interacting three-dimensional insulators. I will illustrate this by the example of a multi-orbital tight-binding model with chiral symmetry that supports a dispersionless band of zero-energy modes with non-trivial density operator algebra. Finally, the possible consequences for strongly correlated electronic states in three dimensions will be discussed within a single-mode approximation. |
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