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Evidence for supercritical behaviour of high-pressure liquid hydrogen


Hydrogen, the simplest and most abundant element in the Universe, develops a remarkably complex behaviour upon compression1. Since Wigner predicted the dissociation and metallization of solid hydrogen at megabar pressures almost a century ago2, several efforts have been made to explain the many unusual properties of dense hydrogen, including a rich and poorly understood solid polymorphism1,3,4,5, an anomalous melting line6 and the possible transition to a superconducting state7. Experiments at such extreme conditions are challenging and often lead to hard-to-interpret and controversial observations, whereas theoretical investigations are constrained by the huge computational cost of sufficiently accurate quantum mechanical calculations. Here we present a theoretical study of the phase diagram of dense hydrogen that uses machine learning to ‘learn’ potential-energy surfaces and interatomic forces from reference calculations and then predict them at low computational cost, overcoming length- and timescale limitations. We reproduce both the re-entrant melting behaviour and the polymorphism of the solid phase. Simulations using our machine-learning-based potentials provide evidence for a continuous molecular-to-atomic transition in the liquid, with no first-order transition observed above the melting line. This suggests a smooth transition between insulating and metallic layers in giant gas planets, and reconciles existing discrepancies between experiments as a manifestation of supercritical behaviour.

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Fig. 1: Thermodynamic properties of high-pressure hydrogen predicted by the MLP based on PBE DFT.
Fig. 2: Polyamorphic solution model fits of the high-pressure hydrogen system.

Data availability

The data supporting the findings of this study are available within the paper, and all input files that are necessary to reproduce the reported results are included in Supplementary Information. All data generated for the study are available upon request from the corresponding author, and the MLP for hydrogen constructed here are available at


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We are thankful to G. Ackland, H. Geng. and R. Redmer, who shared their AIMD trajectories for us to benchmark the MLP. We thank S. Sorella for providing the VMC training dataset. We acknowledge D. Frenkel, B. Monserrat, M. Casula, A. M. Saitta, R. Helled, G. Carleo and S. Sorella for discussions. B.C. acknowledges funding from the Swiss National Science Foundation (project P2ELP2-184408), resources provided by the Cambridge Tier-2 system funded by EPSRC Tier-2 capital grant EP/P020259/1 and by CSCS under project ID s957. G.M. acknowledges financial support from the Swiss National Science Foundation through grant number 200021-179312. C.J.P. is supported by the Royal Society through a Royal Society Wolfson Research Merit award and the EPSRC through grant EP/P022596/1. M.C. acknowledges funding from the Swiss National Science Foundation (project 200021-182057).

Author information




B.C., G.M. and M.C. conceptualized the research; B.C., C.J.P. and M.C. performed the research and analysed the data; B.C., G.M., C.J.P. and M.C. wrote the paper.

Corresponding author

Correspondence to Bingqing Cheng.

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The authors declare no competing interests.

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Peer review information Nature thanks Graeme Ackland and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

This file contains Supplementary Methods, a detailed description of results not reported in the main text and additional analysis. It contains Supplementary Figures 1-19.

Supplementary Data

This zipped folder contains three machine learning potentials for high pressure hydrogen based on PBE DFT, BLYP DFT and VMC, as well as all necessary simulation input files.

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Cheng, B., Mazzola, G., Pickard, C.J. et al. Evidence for supercritical behaviour of high-pressure liquid hydrogen. Nature 585, 217–220 (2020).

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