Noncommutative geometry for three-dimensional topological insulators

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.

Time-reversal symmetric hierarchy of fractional incompressible liquids

“…The stability of this edge theory to the (timereversal symmetric) breaking of translation invariance along the boundary (including the breaking of the spin conservation law associated to the spin vector S) was studied in Ref. 44. The correlation functions for the Fermi-Bose excitations along the edge were computed and shown to be a product over the functions (3.30a).…”

“…III the correspondence between the bulk theory and the edge theory whose stability to the breaking of translation invariance and residual spin-1/2 U(1) symmetry was studied in Ref. 44. Finally, we shall generalize in Sec.…”

“…Past experience with the FQHE has thus guided recent attempts to either construct timereversal symmetric edge theories or to construct timereversal symmetric bulk wave functions supporting local excitations carrying fractional quantum numbers. 37,[40][41][42][43][44][45] While numerical support for a time-reversal symmetric topological phase of matter was given by Neupert et al in their study of a lattice model for interacting itinerant electrons, 44 a description in terms of an effective theory is desirable to reveal the universal properties of such a phase. In Ref.…”

“…In Ref. 44, the universal properties such as the topological degeneracies of the ground state manifold were explored with the help of a family of edge theories. In this paper, we are going to construct the corresponding bulk topological theory by generalizing the hierarchy of Abelian FQHEs to the hierarchy of Abelian fractional quantum spin Hall effects (FQSHEs) in Sec.…”

Elementary formula for the Hall conductivity of interacting systems

“…23,24 Indeed, under those basis transformations of the single-particle Hilbert space that are diagonal in momentum, i.e., those that commute with any P Q , the operator A transforms like an operator-valued gauge field. Explicit representations of the operators T and A will be given in Eqs.…”