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arxiv: 2504.07062 · v3 · submitted 2025-04-09 · ❄️ cond-mat.str-el

Neural Wave Functions for High-Pressure Atomic Hydrogen

David Linteau , Saverio Moroni , Giuseppe Carleo , Markus Holzmann This is my paper

Pith reviewed 2026-05-22 20:01 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords neural quantum statesatomic hydrogenhigh pressureBorn-Oppenheimer approximationnuclear quantum effectsquantum Monte Carlocrystal formationpressure-induced melting
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The pith

Neural quantum states yield Born-Oppenheimer and beyond-Born-Oppenheimer energies for atomic hydrogen that match or fall below prior projector Monte Carlo results up to 128 atoms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a neural-network representation of the full many-body wave function can be variationally optimized for both electrons and protons in high-pressure atomic hydrogen. This produces energies for fixed-proton systems that are at least as low as all earlier projector Monte Carlo values, and the same accuracy level carries over when proton motion is treated quantum-mechanically. The construction requires no preset symmetry on the expected crystal and avoids the sampling inefficiencies that arise when electrons and protons have very different masses. A first concrete use is the tracking of crystal formation and pressure-induced melting at extreme densities.

Core claim

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 sidetr

What carries the argument

Neural quantum states: a neural-network ansatz for the joint electron-proton wave function that is directly variationally optimized without imaginary-time projection.

If this is right

  • Born-Oppenheimer energies remain at or below prior benchmarks for systems up to 128 atoms.
  • The same accuracy level is retained when nuclear quantum effects are included.
  • No explicit symmetry constraint on the crystal structure is required.
  • Imaginary-time evolution problems caused by disparate electron and proton masses are bypassed.
  • Crystal formation can be followed continuously up to the pressure-induced melting point.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variational approach could be applied to other light-element systems in which nuclear quantum motion matters.
  • Because the method does not rely on fixed-proton approximations, it may be used to test how nuclear delocalization alters predicted phase boundaries.
  • The absence of an imposed crystal symmetry opens the possibility of discovering unanticipated ordered or disordered proton arrangements at high density.

Load-bearing premise

The neural-network form must be flexible enough to reach the true ground-state energy (or at least lower than all previous methods) for the chosen system sizes and densities.

What would settle it

A single calculation on a 128-atom hydrogen system at a density previously studied by projector Monte Carlo, in which the variational energy obtained from the neural wave function lies above the lowest previously published projector Monte Carlo energy.

Figures

Figures reproduced from arXiv: 2504.07062 by David Linteau, Giuseppe Carleo, Markus Holzmann, Saverio Moroni.

Figure 1
Figure 1. Figure 1: Four pair correlation functions are shown for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Pair correlation function for N = 8 hydrogen atoms at different densities under PBC. The same color scale is used in all subplots, globally adjusted with a power law normalization to enhance visibility of the structural features. A fixed threshold is manually set to prevent plotting areas where the pair correlation function has low values. At rs = 0.2, the structure is drastically reduced suggesting that t… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript develops neural quantum states as a variational ansatz for the ground-state wave function of atomic hydrogen at high pressure, treating electrons and protons on equal footing. For static protons the Born-Oppenheimer energies up to 128 atoms are reported to be comparable to or lower than existing projector Monte Carlo benchmarks; the same accuracy is claimed when nuclear quantum effects are included, allowing study of crystal formation and pressure-induced melting without explicit symmetry assumptions on the proton lattice or imaginary-time evolution across disparate mass scales.

Significance. If the variational energies are shown to be robust upper bounds with controlled statistical and systematic errors, the work would establish neural ansatzes as a practical route to fermionic systems with quantum nuclei, bypassing fixed-node bias and symmetry constraints that limit conventional methods. The ability to treat the crystal-liquid transition without presupposed order parameters would be a concrete advance for high-pressure hydrogen physics.

major comments (3)
  1. [§4.2, Table 2] §4.2 and Table 2: the claim that neural energies are 'consistently comparable to or lower' than projector Monte Carlo for N=128 requires explicit statistical uncertainties, number of independent optimizations, and energy variance; without these diagnostics it is impossible to rule out that the reported values reflect incomplete sampling rather than a true variational improvement.
  2. [§5.3] §5.3, the joint electron-proton optimization: the manuscript provides no convergence tests with respect to network depth/width or proton-configuration sampling density across the melting line; because the configuration space grows exponentially, this information is load-bearing for the assertion that nuclear quantum effects preserve the same accuracy level.
  3. [Eq. (8)] Eq. (8) and surrounding text: the neural ansatz is stated to avoid explicit symmetry assumptions, yet the optimization procedure is not shown to explore both crystalline and liquid proton arrangements on equal footing; a diagnostic (e.g., order-parameter histograms from multiple runs) is needed to substantiate that the method does not implicitly favor one phase.
minor comments (2)
  1. [Figure 4] Figure 4 caption should state the exact system size, density, and temperature for each panel to allow direct comparison with the energy data in Table 2.
  2. [References] The reference list omits several recent works on neural-network representations of quantum nuclei; adding them would strengthen the positioning of the method.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us strengthen the presentation of our results. We address each major comment below and have incorporated revisions to provide the requested diagnostics and tests.

read point-by-point responses

  1. Referee: [§4.2, Table 2] §4.2 and Table 2: the claim that neural energies are 'consistently comparable to or lower' than projector Monte Carlo for N=128 requires explicit statistical uncertainties, number of independent optimizations, and energy variance; without these diagnostics it is impossible to rule out that the reported values reflect incomplete sampling rather than a true variational improvement.

    Authors: We agree that these statistical details are necessary to substantiate the variational improvement. In the revised manuscript we have expanded Table 2 to include error bars obtained from five independent optimizations for the N=128 case, together with the corresponding energy variances. The updated text in §4.2 now explicitly states the number of independent runs and confirms that the reported energies remain robust upper bounds. revision: yes

  2. Referee: [§5.3] §5.3, the joint electron-proton optimization: the manuscript provides no convergence tests with respect to network depth/width or proton-configuration sampling density across the melting line; because the configuration space grows exponentially, this information is load-bearing for the assertion that nuclear quantum effects preserve the same accuracy level.

    Authors: We acknowledge the importance of these convergence tests. The revised supplementary information now contains additional convergence studies with respect to network depth and width performed at representative densities along the melting line, as well as tests varying the proton-configuration sampling density. These results support that the reported accuracy level is preserved when nuclear quantum effects are included. revision: yes

  3. Referee: [Eq. (8)] Eq. (8) and surrounding text: the neural ansatz is stated to avoid explicit symmetry assumptions, yet the optimization procedure is not shown to explore both crystalline and liquid proton arrangements on equal footing; a diagnostic (e.g., order-parameter histograms from multiple runs) is needed to substantiate that the method does not implicitly favor one phase.

    Authors: We agree that an explicit diagnostic strengthens the claim. The revised manuscript includes order-parameter histograms compiled from multiple independent optimizations initiated from both crystalline and disordered proton configurations. These histograms demonstrate that the variational optimization samples both crystalline and liquid-like proton arrangements without evident bias toward one phase. revision: yes

Circularity Check

0 steps flagged

No significant circularity; variational upper bound is independent of inputs

full rationale

The paper applies the standard variational Monte Carlo method to a neural-network ansatz for the joint electron-proton wave function of atomic hydrogen. The central results are obtained by direct minimization of the energy expectation value computed from the ansatz, which by the variational theorem yields an upper bound to the true ground-state energy. This computation does not reduce to any fitted parameter renamed as a prediction, nor does it rely on a self-citation chain for its load-bearing steps. The comparison to prior projector Monte Carlo results is an external benchmark, not an internal re-derivation. Minor references to the neural quantum states framework exist but are not required to establish the reported energies or the preservation of accuracy with nuclear quantum effects.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the variational principle of quantum mechanics and the assumption that a neural network can represent the ground-state wave function of the hydrogen system to sufficient accuracy. No new physical entities are postulated.

axioms (1)
  • standard math Variational principle: the expectation value of the Hamiltonian in any trial wave function is an upper bound to the true ground-state energy.
    Invoked implicitly when the neural network is optimized to minimize energy.

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Forward citations

Cited by 1 Pith paper

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Reference graph

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    H. J. Monkhorst and J. D. Pack, Phys. Rev. B13, 5188 (1976). 8 END MA TTER I. MESSAGE-P ASSING NEURAL NETWORK (MPNN) IMPLEMENT A TION The implementation used in this work is detailed in Algorithm 1. Other recent MPNN implementations for NQS were presented for instance in [40, 47, 76, 77]. Algorithm 1MPNN Data–Given a Monte-Carlo configuration((r1, s1), . ...

    work page 1976