  Gauge theories with matter fields in various representations play an
important role in different branches of physics. Recently, it was proposed that
several aspects of the interesting pseudogap phase of cuprate superconductors
near optimal doping may be explained by an emergent $SU(2)$ gauge symmetry.
Around the transition with positive hole-doping, one can construct a
$(2+1)-$dimensional $SU(2)$ gauge theory coupled to four adjoint scalar fields
which gives rise to a rich phase diagram with a myriad of phases having
different broken symmetries. We study the phase diagram of this model on the
Euclidean lattice using the Hybrid Monte Carlo algorithm. We find the existence
of multiple broken phases as predicted by previous mean field studies.
Depending on the quartic couplings, the $SU(2)$ gauge symmetry is broken down
either to $U(1)$ or $\mathbb{Z}_2$ in the perturbative description of the
model. We further study the confinement-deconfinement transition in this
theory, and find that both the broken phases are deconfining in the range of
volumes that we studied. However, there exists a marked difference in the
behavior of the Polyakov loop between the two phases.
