Solvable Lattice Hamiltonians with Fracitonal Hall Conductivity

We construct a class of lattice Hamiltonians -- both bosonic and fermionic ones -- that exhibit fractional Hall conductivity. These Hamiltonians, while not being exactly solvable, can be controllably solved in their low energy sectors, through a combination of perturbative and exact techniques. Our construction demonstrates a systematic way to circumvent the Kapustin-Fidkowski no-go theorem. More broadly, our work may shed light on the general study of the symmetry enriched topological phases which have gapless edges protected by symmetry.