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  • Letter
  • Open Access

Neural wave functions for high-pressure atomic hydrogen

David Linteau1,2,*, Saverio Moroni3, Giuseppe Carleo1,2, and Markus Holzmann4

  • *Contact author: david.linteau@epfl.ch
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Phys. Rev. Research 8, L022007 – Published 7 April, 2026

DOI: https://doi.org/10.1103/t72h-tkcx

Abstract

We leverage the power of neural quantum states to describe the ground-state wave function of solid and liquid atomic hydrogen, including both electronic and protonic degrees of freedom. For static protons, the resulting Born-Oppenheimer energies are consistently comparable to or lower than all previous projector Monte Carlo results for systems containing up to 128 hydrogen atoms. The same level of accuracy is preserved upon inclusion of nuclear quantum effects, thus going beyond the Born-Oppenheimer approximation. In addition, our description overcomes major limitations of current wave functions, notably by avoiding any explicit symmetry assumption on the expected quantum crystal and sidestepping efficiency issues of imaginary time evolution with disparate mass scales. As a first application, we examine crystal formation in an extremely high-density region up to pressure-induced melting.

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Phys. Rev. Research 8, L022007– Published 7 April, 2026
  • Received 16 April 2025
  • Revised 16 November 2025
  • Accepted 4 February 2026
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