High-pressure superconducting state in hydrogen
  The paper determines the thermodynamic parameters of the superconducting state in the metallic atomic hydrogen under the pressure at $1$ TPa, $1.5$ TPa, and $2.5$ TPa. The calculations were conducted in the framework of the Eliashberg formalism. It has been shown that the critical temperature is very high (in the range from $301.2$ K to $437.3$ K), as well as high are the values of the electron effective mass (from $3.43$ $m_{e}$ to $6.88$ $m_{e}$), where $m_{e}$ denotes the electron band mass. The ratio of the low-temperature energy gap to the critical temperature explicitly violates the predictions of the BCS theory: $2\Delta\left(0\right)/k_{B}T_{C}\in\left<4.84,5.85\right>$. Additionally, the free energy difference between the superconducting and normal state, the thermodynamic critical field, and the specific heat of the superconducting state have been determined. Due to the significant strong-coupling and retardation effects those quantities cannot be correctly described in the framework of the BCS theory. 
A. M. Duda, R. Szcz\k{e}\'sniak, M. A. Sowi\'nska, A. H. Kosiacka
cond-mat.supr-con