Superconductivity in metallic hydrogen

<h2>Summary</h2> Superconductivity, the lossless flow of electric current, occurs typically at very low temperatures. A possible exception is highly pressurized hydrogen, for which room temperature superconductivity has been predicted. However, as a result of various approximations used, conflicting theoretical predictions exist for the temperatures where superconductivity is expected to occur in highly pressurized hydrogen. Here, we avoid those approximations and exploit the "jellium" model proposed in 1966 by De Gennes, where superconductivity involves the combination of Coulomb repulsion between the electrons and Coulomb attraction between the protons and the electrons. We confirm that metallic hydrogen should indeed exhibit superconductivity, but this is limited to temperatures far below previous estimates. We also find that the superconducting order develops over an energy range significantly exceeding the characteristic phonon energy and that the phase of the order parameter flips 180° at the characteristic phonon energy above and below the Fermi energy.