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 (QED4) to QED4 with fractionally charged excitations, the fractional S-duality. Such QED4 can be obtained by gauging the 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 is an element of Z). Here, 1/t specifies the minimal electric charge carried by bulk excitations. Such states with time-reversal and 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 is an element of Z(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 is an element of Zeven) cannot be realized through any stacking. Finally, we study the surface topological order of fractional topological insulators.