  We discuss the role of dynamical many-electron effects in the physics of iron
and iron-rich solid alloys under applied pressure on the basis of recent ab
initio studies employing the dynamical mean-field theory (DMFT). Electronic
correlations in iron in the moderate pressure range up to 60 GPa are discussed
in the first section. DMFT-based methods predict an enhancement of electronic
correlations at the pressure-induced transition from body-centered cubic (bcc)
alpha-Fe to hexagonal close-packed (hcp) epsilon-Fe. In particular, the
electronic effective mass, scattering rate and electron-electron contribution
to the electrical resistivity undergo a step-wise increase at the transition
point. One also finds a significant many-body correction to the epsilon-Fe
equation of state, thus clarifying the origin of discrepancies between previous
DFT studies and experiment. An electronic topological transition is predicted
to be induced in epsilon-Fe by many-electron effects; its experimental
signatures are analyzed. Next section focuses on the geophysically relevant
pressure-temperature regime of the Earth's inner core (EIC) corresponding to
the extreme pressure of 360 GPa combined with temperatures up to 6000 K. The
three iron allotropes (bcc, hcp and face-centered-cubic) previously proposed as
possible stable phases at such conditions are found to exhibit qualitatively
different many-electron effects as evidenced by a strongly non-Fermi-liquid
metallic state of bcc-Fe and an almost perfect Fermi liquid in the case of
hcp-Fe. A recent active discussion on the electronic state and transport
properties of hcp-Fe at the EIC conditions is reviewed in details. We also
discuss the impact of a Ni admixture, which is expected to be present in the
core matter. We conclude by outlining some limitation of the present DMFT-based
framework and perspective directions for further development.
