Fractional $S$-duality, classification of fractional topological insulators, and surface topological order

In this paper, we propose a generalization of the $S$-duality of four-dimensional quantum electrodynamics (${\text{QED}}_4$) to ${\text{QED}}_4$ with fractionally charged excitations, the fractional $S$-duality. Such ${\text{QED}}_4$ can be obtained by gauging the $\text{U(1)}$ symmetry of a topologically ordered state with fractional charges. When time-reversal symmetry is imposed, the axion angle ($\theta$) can take a nontrivial but still time-reversal-invariant value $\pi /t^2$ ($t\in \mathbb{Z}$). Here, $1/t$ specifies the minimal electric charge carried by bulk excitations. Such states with time-reversal and $\text{U(1)}$ global symmetry (fermion number conservation) are fractional topological insulators (FTIs). We propose a topological quantum field theory description, which microscopically justifies the fractional $S$-duality. Then, we consider stacking operations (i.e., a direct sum of Hamiltonians) among FTIs. We find that there are two topologically distinct classes of FTIs: type I and type II. Type I ($t\in {\mathbb{Z}}_{\mathrm{odd}}$) can be obtained by directly stacking a noninteracting topological insulator and a fractionalized gapped fermionic state with minimal charge $1/t$ and vanishing $\theta$. But type II ($t\in {\mathbb{Z}}_{\mathrm{even}}$) cannot be realized through any stacking. Finally, we study the surface topological order of fractional topological insulators.