  We study the ground state and the thermal phase diagram of a two-species
Bose-Hubbard model, with $U(1)\times \mathbb{Z}_2$ symmetry, describing atoms
and molecules on a 2D optical lattice interacting via a Feshbach resonance.
Using quantum Monte Carlo simulations and mean field theory, we show that the
conversion between the two-species, coherently coupling the atomic and
molecular states, has a crucial impact on the Mott-Superfluid transition and
stabilizes an insulating phase with a gap controlled by the conversion term --
\textit{the Feshbach insulator} -- instead of a standard Mott insulating phase.
Depending on the detuning between atoms and molecules, this model exhibits
three phases: the Feshbach insulator, a molecular condensate coexisting with
non condensed atoms and a mixed atomic-molecular condensate. Employing a
finite-size scaling method, we observe 3D XY (3D Ising) transition when $U(1)$
($ \mathbb{Z}_2$) is broken whereas the transition is first-order when both
$U(1)$ and $ \mathbb{Z}_2$ symmetries are spontaneously broken. The finite
temperature phase diagram is also discussed. The thermal disappearance of the
molecular superfluid leads to a Berezinskii-Kosterlitz-Thouless transition with
unusual universal jump in the superfluid density. The loss of the
quasi-long-range coherence of the mixed atomic and molecular superfluid is more
subtle since only atoms exhibit conventional Berezinskii-Kosterlitz-Thouless
criticality. We also observe a classical first-order transition between the
mixed superfluid and the normal Bose liquid at low temperature.
