Topological BF field theory description of topological insulators

TL;DR

This paper proposes a topological BF field theory framework to describe topological insulators in two and three dimensions, unifying their properties and surface states, and suggesting potential for fractional topological phases.

Contribution

It introduces a BF theory description for 2D and 3D topological insulators, connecting them to known theories and exploring fractional phases.

Findings

BF theory describes 2D topological insulators as equivalent to Chern-Simons theories.

3D topological insulators exhibit gapless surface states and axion electrodynamics within the BF framework.

Potential for describing fractional topological insulators with fractional statistics.

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.