  Abelian Chern-Simons-Maxwell theory can emerge from the bosonisation of the
2+1-dimensional Thirring model that describes interacting Dirac fermions. Here
we show how the Thirring model manifests itself in the low energy limit of a
two-dimensional tight-binding model of spinless fermions. To establish that we
employ a modification of Haldane's model, where the "doubling" of fermions is
rectified by adiabatic elimination. Subsequently, fermionic interactions are
introduced that lead to the analytically tractable Thirring model. By local
density measurements of the lattice fermions we can establish that for specific
values of the couplings the model exhibits the confining 2+1-dimensional QED
phase or a topological ordered phase that corresponds to the Chern-Simons
theory. The implementation of the model as well as the measurement protocol are
accessible with current technology of cold atoms in optical lattices.
