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.Comment: 18 page

Chiral gauge field and axial anomaly in a Weyl semimetal

“…Based on both low energy effective theory approach and numerical results in a microscopic model of doped topological insulators, we propose two physical consequences of the anomaly, the chiral modes in magnetic vortices, and the magnon-plasmon coupling. An open question is whether it is possible to write down a topological effective field theory to characterize the transport properties of the Weyl semimetals, as in the topological insulators 5,16,37 . Since there are gapless fermions in Weyl semimetals, it is not clear whether a local bosonic effective field theory can be obtained.…”

“…By dimensional reduction, the (4+1)-d topological insulator is reduced to the (3+1)-d TI [8][9][10] and the Weyl fermion is reduced to (2+1)-d surface states of the TI. Weyl fermions also appear directly in (3 + 1)-d gapless electron systems, which are named as "Weyl semi-metals" 4,[11][12][13][14][15][16][17][18][19][20][21] . Since a system with both TR and parity (P) symmetries have all energy bands doubly degenerate, the Weyl semi-metal state can only be realized in a system breaking TR and/or P symmetry.…”

D-algebra structure of topological insulators

“…The D-commutator hints at a different group structure from the usual gauge theories, such as higher gauge theories [27,28]. In light of this, the recent proposal [29] to describe topological insulators by a BF theory [30] looks very promising. In two dimensions, the classical limit of the GMP algebra is isomorphic to the algebra of area preserving diffeomorphisms, and is related to incompressibility.…”

Three-dimensional topological insulators and bosonization

“…The topological phases of matter [1][2][3][4] have been described by several approaches, such as wavefunction modeling [5], band theory [6][7][8][9][10] and effective field theory of boundary excitations [11,12], whose interplay has been extremely rich and fruitful.…”