Index theorem, spin Chern Simons theory and fractional magnetoelectric effect in strongly correlated topological insulators

Abstract: Making use of index theorem and spin Chern Simons theory, we construct an effective topological field theory of strongly correlated topological insulators coupling to a nonabelian gauge field $ SU(N) $ with an interaction constant $ g $ in the absence of the time-reversal symmetry breaking. If $ N $ and $ g $ allow us to define a t'Hooft parameter $ \lambda $ of effective coupling as $ \lambda = N g^{2} $, then our construction leads to the fractional quantum Hall effect on the surface with Hall conductance $ \sigma_{H}^{s} = \frac{1}{4\lambda} \frac{e^{2}}{h} $. For the magnetoelectric response described by a bulk axion angle $ \theta $, we propose that the fractional magnetoelectric effect can be realized in gapped time reversal invariant topological insulators of strongly correlated bosons or fermions with an effective axion angle $ \theta_{eff} = \frac{\pi}{2 \lambda} $ if they can have fractional excitations and degenerate ground states on topologically nontrivial and oriented spaces. Provided that an effective charge is given by $ e_{eff} = \frac{e}{\sqrt{2 \lambda}} $, it is shown that $ \sigma_{H}^{s} = \frac{e_{eff}^{2}}{2h} $, resulting in a surface Hall conductance of gapless fermions with $ e_{eff} $ and a pure axion angle $ \theta = \pi $.