  Novel two-dimensional (2D) emergent fermions and negative Poisson's ratio in
2D materials are fascinating subjects of research. Here, based on
first-principles calculations and theoretical analysis, we predict that the
hexacoordinated Mg$_{2}$C monolayer hosts both exotic properties. We analyze
its phonon spectrum, reveal the Raman active modes, and show that it has small
in-plane stiffness constants. Particularly, under the tensile strain in the
zigzag direction, the Mg$_{2}$C monolayer shows an intrinsic negative Poisson's
ratio $\sim -0.023$, stemming from its unique puckered hinge structure. The
material is metallic at its equilibrium state. A moderate biaxial strain can
induce a metal-semimetal-semiconductor phase transition, during which several
novel types of 2D fermions emerge, including the anisotropic Dirac fermions
around 12 tilted Dirac points in the metallic phase, the $2$D double Weyl
fermions in the semimetal phase where the conduction and valence bands touch
quadratically at a single Fermi point, and the 2D pseudospin-1 fermions at the
critical point of the semimetal-semiconductor phase transition where three
bands cross at a single point on the Fermi level. In addition, uniaxial strains
along the high-symmetry directions break the three-fold rotational symmetry and
reduce the number of Dirac points. Interestingly, it also generates 2D type-II
Dirac points. We construct effective models to characterize the properties of
these novel fermions. Our result reveals Mg$_{2}$C monolayer as an intriguing
platform for the study of novel 2D fermions, and also suggests its great
potential for nanoscale device applications.
