  The pressure-induced phase transition, elasticity behavior, thermodynamic
properties, and $P\mathtt{-}T$ phase diagram of $\alpha$, $\omega$, and $\beta$
equiatomic HfTi alloy are investigated using first-principles
density-functional theory (DFT). The simulated pressure-induced phase
transition of the alloy follows the sequence of
$\alpha\mathtt{\rightarrow}\omega\mathtt{\rightarrow}\beta$, in agreement with
the experimental results of Hf and Ti metals. Our calculated elastic constants
show that the $\alpha$ and $\omega$ phases are mechanically stable at ambient
pressure, while the $\beta$ phase is unstable, where a critical pressure of
18.5 GPa is predicted for its mechanical stability. All the elastic constants,
bulk modulus, and shear modulus increase upon compression for the three phases
of HfTi. The ductility of the alloy is shown to be well improved with respect
to pure Hf and Ti metals. The Mulliken charge population analysis illustrates
that the increase of the d-band occupancy will stabilize the $\beta$ phase
under pressure. The phonon spectra and phonon density of states are studied
using the supercell approach for the three phases, and the stable nature of
$\alpha$ and $\omega$ phases at ambient pressure are observed, while the
$\beta$ phase is only stable along the [110] direction. With the Gibbs free
energy calculated from DFT-parametrized Debye model as a function of
temperature and pressure, the phase transformation boundaries of the $\alpha$,
$\omega$, and $\beta$ phases of HfTi are identified.
