  By ungauging a recently discovered lattice rotor model for Chern-Simons
theory, we create an exactly soluble path integral on spacetime lattice for
$U^\kappa(1)$ Symmetry Protected Topological (SPT) phases in $2+1$ dimensions
with a non-zero Hall conductance. We then convert the path integral on a $2+1$d
spacetime lattice into a $2$d Hamiltonian lattice model, and show that the
Hamiltonian consists of mutually commuting local projectors. We confirm the
non-zero Hall conductance by calculating the Chern number of the exact ground
state. It has recently been suggested that no commuting projector model can
host a nonzero Hall conductance. We evade this no-go theorem by considering a
rotor model, with a countably infinite number of states per site.
