Hydrogen is the most abundant element in the universe, and its properties under conditions of high temperature and pressure are crucial to understand the interior of of large gaseous planets and other astrophysical bodies. At ultra high pressures solid hydrogen has been predicted to transform into a quantum fluid, because of its high zero point motion. Here we report first principles two phase coexistence and Z method determinations of the melting line of solid hydrogen in a pressure range spanning from 30 to 600 GPa. Our results suggest that the melting line of solid hydrogen, as derived from classical molecular dynamics simulations, reaches a minimum of 367 K at about 430 GPa, at higher pressures the melting line of the atomics Cs IV phase regain a positive slope. In view of the possible importance of quantum effects in hydrogen at such low temperatures, we also determined the melting temperature of the atomic CsIV phase at pressures of 400, 500, 600 GPa, employing Feynman path integral simulations. These result in a downward shift of the classical melting line by about 100 K, and hint at a possible secondary maximum in the melting line in the region between 500 and 600 GPa, testifying to the importance of quantum effects in this system. Combined, our results imply that the stability field of the zero temperature quantum liquid phase, if it exists at all, would only occur at higher pressures than previously thought.
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