  We investigate a model of interacting Dirac fermions in $2+1$ dimensions with
$M$ flavors and $N$ colors having the $\mathrm{U}(M)\times \mathrm{SU}(N)$
symmetry. In the large-$N$ limit, we find that the $\mathrm{U}(M)$ symmetry is
spontaneously broken in a variety of ways. In the vacuum, when the
parity-breaking flavor-singlet mass is varied, the ground state undergoes a
sequence of $M$ first-order phase transitions, experiencing $M+1$ phases
characterized by symmetry breaking $\mathrm{U}(M)\to \mathrm{U}(M-k)\times
\mathrm{U}(k)$ with $k\in\{0,1,2,\cdots,M\}$, bearing a close resemblance to
the vacuum structure of three-dimensional QCD. At finite temperature and
chemical potential, a rich phase diagram with first and second-order phase
transitions and tricritical points is observed. Also exotic phases with
spontaneous symmetry breaking of the form as $\mathrm{U}(3)\to
\mathrm{U}(1)^3$, $\mathrm{U}(4)\to \mathrm{U}(2)\times \mathrm{U}(1)^2$, and
$\mathrm{U}(5)\to \mathrm{U}(2)^2\times \mathrm{U}(1)$ exist. For a large
flavor-singlet mass, the increase of the chemical potential $\mu$ brings about
$M$ consecutive first-order transitions that separate the low-$\mu$ phase
diagram with vanishing fermion density from the high-$\mu$ region with a high
fermion density.
