  We report on an experimental study of the Lipkin-Meshkov-Glick model of
quantum spins interacting at infinite range in a transverse magnetic field,
which exhibits a ferromagnetic phase transition in the thermodynamic limit. We
use Dysprosium atoms of electronic spin $J=8$, subjected to a quadratic Zeeman
light shift, to simulate $2J=16$ interacting spins $1/2$. We probe the system
microscopically using single magnetic sublevel resolution, giving access to the
spin projection parity, which is the collective observable characterizing the
underlying $\mathbb{Z}_2$ symmetry. We measure the thermodynamic properties and
dynamical response of the system, and study the quantum critical behavior
around the transition point. In the ferromagnetic phase, we achieve coherent
tunneling between symmetry-broken states, and test the link between symmetry
breaking and the appearance of a finite order parameter.
