  Motivated by the recent synthesis of two-dimensional monolayer AlSb, we
theoretically investigate its ground state and electronic properties using the
first-principles calculations coupled with Bethe-Salpeter equation. An
excitonic instability is revealed as a result of larger exciton binding energy
than the corresponding one-electron energy gap by $\sim$0.1 eV, which is an
indicative of a many-body ground state accompanied by spontaneous exciton
generation. Including the spin-orbit coupling is proven to be a must to
correctly predict the ground state. At room temperature, the two-dimensional
monolayer AlSb is a direct gap semiconductor with phonon-limited electron and
hole mobilities both around 1700 cm$^2$/V$\cdot$s. These results show that
monolayer AlSb may provide a promising platform for realization of the
excitonic insulator and for applications in the next-generation electronic
devices.
