  Superionic ices with highly mobile protons within the stable oxygen
sub-lattice occupy an important proportion of the phase diagram of ice and
widely exist in the interior of icy giants and throughout the universe.
Understanding the thermal transport in superionic ice is vital for the thermal
evolution of icy planets. However, it is highly challenging due to the extreme
thermodynamic conditions and dynamical nature of protons, beyond the capability
of the traditional lattice dynamics and empirical potential molecular dynamics
approaches. In this work, by utilizing the deep potential molecular dynamics
approach, we investigate the thermal conductivity of ice-VII and superionic
ice-VII" along the isobar of $p = 30\ \rm{GPa}$. A non-monotonic trend of
thermal conductivity with elevated temperature is observed. Through heat flux
decomposition and trajectory-based spectra analysis, we show that the
thermally-activated proton diffusion in ice-VII and superionic ice-VII"
contribute significantly to heat convection, while the broadening in
vibrational energy peaks and significant softening of transverse acoustic
branches lead to a reduction in heat conduction. The competition between proton
diffusion and phonon scattering results in anomalous thermal transport across
the superionic transition in ice. This work unravels the important role of
proton diffusion in the thermal transport of high-pressure ice. Our approach
provides new insights into modeling the thermal transport and atomistic
dynamics in superionic materials.
