Topological BF field theory description of topological insulators

Abstract

Topological phases of matter are described universally by topological field theories in the same way that symmetry-breaking phases of matter are described by Landau-Ginzburg field theories. We propose that topological insulators in two and three dimensions are described by a version of abelian $BF$ theory. For the two-dimensional topological insulator or quantum spin Hall state, this description is essentially equivalent to a pair of Chern-Simons theories, consistent with the realization of this phase as paired integer quantum Hall effect states. The $BF$ description can be motivated from the local excitations produced when a $\pi$ flux is threaded through this state. For the three-dimensional topological insulator, the $BF$ description is less obvious but quite versatile: it contains a gapless surface Dirac fermion when time-reversal-symmetry is preserved and yields "axion electrodynamics", i.e., an electromagnetic $E\cdot B$ term, when time-reversal symmetry is broken and the surfaces are gapped. Just as changing the coefficients and charges of 2D Chern-Simons theory allows one to obtain fractional quantum Hall states starting from integer states, $BF$ theory could also describe (at a macroscopic level) fractional 3D topological insulators with fractional statistics of point-like and line-like objects.

18 pages

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Cited by in corpus (115)

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• Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries
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• Noncommutative geometry for three-dimensional topological insulators
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• Effective field theories for topological insulators by functional bosonization
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• Tensor Berry connections and their topological invariants
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• Models of three-dimensional fractional topological insulators
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• A new Hamiltonian for the Topological BF phase with spinor networks
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• Gapless edge states of BF field theory and translation-symmetric Z2 spin liquids
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• Higher (Odd) Dimensional Quantum Hall Effect and Extended Dimensional Hierarchy
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• Holographic reduction of Maxwell-Chern-Simons theory
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