Can a nondegenerate fermionic topological Mott insulator (TMI) state support an emergent bosonic topological order? Based on my recent study and motivated by a recent paper, I have a naive question. 
Consider a 2d Hubbard model for electrons at half filling 
$H=\sum c_k^\dagger h_k c_k+U\sum n_{i\uparrow }n_{i\downarrow }$, assuming the following facts:
(1) the noninteracting part is a Chern insulator (CI)
(2) for large enough $U(>U_c)$ the system is a nonmagnetic Mott insulator and it has a unique ground state
(3) this nonmagnetic Mott insulator has a nonzero Chern number $C=\frac{1}{24\pi^2}\int dk_0dk^2tr(\epsilon^{\mu \nu \lambda }G\partial_\mu G^{-1}G\partial_\nu G^{-1}G\partial_\lambda G^{-1})$ equal to that of the CI, indicating it is a TMI.
On the other hand, in the the slave-rotor mean-field description 
$c_{i\sigma }=e^{i\theta_i}f_{i\sigma }$, the above TMI phase may be interpreted as a fractionalized CI in terms of the spinons $f_{i\sigma }$ or a chiral spin liquid (CSL, the projected spinon mean-field state).
Question: from the slave-rotor viewpoint, the CSL with spinon band Chern number $C$ seems to have a (bosonic) topological degeneracy, is this possible for our fermionic TMI without ground-state degeneracy?
 A: Clearly the "TMI" and the slave-rotor mean-field state are very different, because the TMI, as you assume, has no topological degeneracy while the other state is topologically ordered.
However, I feel this answer is not very meaningful without seeing more details of the slave-rotor mean-field state. I'm afraid this is not a very well-known (or even well-accepted...) result, so perhaps a reference will be helpful. The nature of the "TMI" state is also unclear. If one follows the lines in http://arxiv.org/abs/1510.04278 (and related papers from the same authors), the "TMI" studied there is just a product state, nothing topological at all. If you are interested in 2d fermionic insulator with unique ground state (so a SPT phase), not worrying about filling for the moment, then we know the classification of such states already: with U(1) and broken time-reversal symmetry (since you said the band part has nonzero Chern number), they are classified again by the Chern number even with strong interactions. All "interacting" Chern insulators are adiabatically connected to non-interacting ones.
