  Using hybrid exchange-correlation functional in ab initio density functional
theory calculations, we study magnetic properties and strain effect on the
electronic properties of $\alpha$-graphyne monolayer. We find that a
spontaneous antiferromagnetic (AF) ordering occurs with energy band gap ($\sim$
0.5 eV) in the equilibrated $\alpha$-graphyne. Bi-axial tensile strain enhances
the stability of AF state as well as the staggered spin moment and value of the
energy gap. The antiferromagnetic semiconductor phase is quite robust against
moderate carrier filling with threshold carrier density up to
1.7$\times$10$^{14}$ electrons/cm$^2$ to destabilize the phase. The spontaneous
AF ordering and strain effect in $\alpha$-graphyne can be well described by the
framework of the Hubbard model. Our study shows that it is essential to
consider the electronic correlation effect properly in $\alpha$-graphyne and
may pave an avenue for exploring magnetic ordering in other carbon allotropes
with mixed hybridization of s and p orbitals.
