  Here, we provide a simple Hubbard-like model of spin-$1/2$ fermions that
gives rise to the SU(2) symmetric Thirring model that is equivalent, in the
low-energy limit, to Yang-Mills-Chern-Simons model. First, we identify the
regime that simulates the SU(2) Yang-Mills theory. Then, we suitably extend
this model so that it gives rise to the SU(2) level $k$ Chern-Simons theory
with $k\geq2$ that can support non-Abelian anyons. This is achieved by
introducing multiple fermionic species and modifying the Thirring interactions,
while preserving the SU(2) symmetry. Our proposal provides the means to
theoretically and experimentally probe non-Abelian SU(2) level $k$ topological
phases.
