  Graphene is famous for being a host of 2D Dirac fermions. However, spin-orbit
coupling introduces a small gap, so that graphene is formally a quantum spin
hall insulator. Here we present symmetry-protected 2D Dirac semimetals, which
feature Dirac cones at high-symmetry points that are \emph{not} gapped by
spin-orbit interactions, and exhibit behavior distinct from both graphene and
3D Dirac semimetals. Using a two-site tight-binding model, we construct
representatives of three possible distinct Dirac semimetal phases, and show
that single symmetry-protected Dirac points are impossible in two dimensions.
An essential role is played by the presence of non-symmorphic space group
symmetries. We argue that these symmetries tune the system to the boundary
between a 2D topological and trivial insulator. By breaking the symmetries we
are able to access trivial and topological insulators as well as Weyl semimetal
phases.
