Anyon structure of fractional topological insulating slabs with gapped surfaces

We consider fractional topological insulators (FTI) in three dimensions whose bulk quasiparticle excitations consist of partons coupled with a $Z_{2n+1}$ gauge theory. Their surface states can acquire an excitation energy gap by either breaking time reversal symmetry, charge $U(1)$ symmetry, or preserving both symmetries while introducing additional T-Pfaffian-like surface topological order. We theoretically study the anyon and symmetry structures of these gapped surface states as well as quasi-$(2+1)$D slabs of FTI with distinct opposite gapped surfaces.