  The electronic state and transport properties of hot dense iron are of the
utmost importance to geophysics. Combining the density functional and dynamical
mean field theories we study the impact of electron correlations on electrical
and thermal resistivity of hexagonal close-packed $\epsilon$-Fe at Earth's core
conditions. $\epsilon$-Fe is found to behave as a nearly perfect Fermi liquid.
The quadratic dependence of the scattering rate in Fermi liquids leads to a
modification of the Wiedemann-Franz law with suppression of the thermal
conductivity as compared to the electrical one. This significantly increases
the electron-electron thermal resistivity which is found to be of comparable
magnitude to the electron-phonon one. The implications of this effect on the
dynamics of Earth's core is discussed.
