Filling constraints on fermionic topological order in zero magnetic field

We consider two-dimensional electron systems in zero magnetic field at fractional filling. For such systems a Lieb-Schultz-Mattis theorem applies, forbidding the existence of a trivial insulator. However, the theorem does not distinguish between bosonic and fermionic systems. In this work we argue that in the case of fermionic systems, the topological orders that are compatible with the microscopic constraints are in general different from the bosonic case. We find different results in the cases of even and odd denominator fillings, with even denominator fillings deviating stronger from the bosonic case. Part of our results also hold in three dimensions.