  The Kapustin-Fidkowski no-go theorem forbids $U(1)$ symmetric topological
orders with non-trivial Hall conductivity in (2+1)d from admitting commuting
projector Hamiltonians, where the latter is the paradigmatic method to
construct exactly solvable lattice models for topological orders. Even if a
topological order would intrinsically have admitted commuting projector
Hamiltonians, the theorem forbids so once its interplay with $U(1)$ global
symmetry which generates Hall conductivity is taken into consideration.
Nonetheless, in this work, we show that for all (2+1)d $U(1)$ symmetric abelian
topological orders of such kind, we can construct a lattice Hamiltonian that is
controllably solvable at low energies, even though not "exactly" solvable;
hence, this no-go theorem does not lead to significant difficulty in the
lattice study of these topological orders. Moreover, for the fermionic
topological orders in our construction, we introduce the lattice notion of
spin-c structure -- a concept important in the continuum that has previously
not been adequately introduced in the lattice context.
