Fractional Topological Insulators in Three Dimensions

Topological insulators can be generally defined by a topological field theory with an axion angle $\theta$ of 0 or $\pi$. In this work, we introduce the concept of fractional topological insulator defined by a fractional axion angle and show that it can be consistent with time reversal $T$ invariance if ground state degeneracies are present. The fractional axion angle can be measured experimentally by the quantized fractional bulk magnetoelectric polarization $P_3$, and a “halved” fractional quantum Hall effect on the surface with Hall conductance of the form ${\sigma }_H=\frac{p}{q}\frac{e^2}{2h}$ with $p$, $q$ odd. In the simplest of these states the electron behaves as a bound state of three fractionally charged “quarks” coupled to a deconfined non-Abelian $SU(3)$ “color” gauge field, where the fractional charge of the quarks changes the quantization condition of $P_3$ and allows fractional values consistent with $T$ invariance.

Figure 1

(a) Quark picture of fractional TI with flavor and color degrees of freedom; (b) surface FQHE vs transport measurements [Eq. (7)]; (c) nontrivial vs trivial fractional TI; (d) Witten effect as a probe of bulk topology.