  The archetypal $3d$ Mott insulator hematite, Fe$_2$O$_3$, is one of the basic
oxide components playing an important role in mineralogy of Earth's lower
mantle. Its high pressure-temperature behavior, such as the electronic
properties, equation of state, and phase stability is of fundamental importance
for understanding the properties and evolution of the Earth's interior. Here,
we study the electronic structure, magnetic state, and lattice stability of
Fe$_2$O$_3$ at ultra-high pressures using the density functional plus dynamical
mean-field theory (DFT+DMFT) approach. In the vicinity of a Mott transition,
Fe$_2$O$_3$ is found to exhibit a series of complex electronic, magnetic, and
structural transformations. In particular, it makes a phase transition to a
metal with a post-perovskite crystal structure and site-selective local moments
upon compression above 75 GPa. We show that the site-selective phase transition
is accompanied by a charge disproportionation of Fe ions, with Fe$^{3\pm
\delta}$ and $\delta \sim 0.05$-$0.09$, implying a complex interplay between
electronic correlations and the lattice. Our results suggest that
site-selective local moments in Fe$_2$O$_3$ persist up to ultra-high pressures
of $\sim$200-250 GPa, i.e., sufficiently above the core-mantle boundary. The
latter can have important consequences for understanding of the velocity and
density anomalies in the Earth's lower mantle.
