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 (QED 4 ) to QED 4 with fractionally charged excitations, the fractional S-duality. Such QED 4 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 (θ) can take a nontrivial but still time-reversal invariant value π/t (t ∈ 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 (FTI). 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 ∈ Zodd) can be obtained by directly stacking a non-interacting topological insulator and a fractionalized gapped fermionic state with minimal charge 1/t and vanishing θ. But type-II (t ∈ Zeven) cannot be realized through any stacking. Finally, we study the Surface Topological Order of fractional topological insulators.