Revisiting the Broken Symmetry Phase of Solid Hydrogen: A Neural Network Variational Monte Carlo Study

Shengdu Chai; Chen Lin; Xinyang Dong; Yuqiang Li; Wanli Ouyang; Lei Wang; X. C. Xie

cond-mat.str-elDec 19, 2025

Revisiting the Broken Symmetry Phase of Solid Hydrogen: A Neural Network Variational Monte Carlo Study

Authors: Shengdu Chai, Chen Lin, Xinyang Dong, Yuqiang Li, Wanli Ouyang, Lei Wang, X. C. Xie

Organizations: Interdisciplinary Center for Theoretical Physics and Information Sciences (ICTPIS), Fudan University, Shanghai 200433, China · Shanghai Artificial Intelligence Laboratory, Shanghai 200232, China · Department of Engineering, University of Oxford, Oxford OX1 4BH, UK · Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China · Department of Information Engineering, The Chinese University of Hong Kong, Hong Kong SAR HKG, China · International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China · Hefei National Laboratory, Hefei 230088, China

Abstract

The crystal structure of high-pressure solid hydrogen remains a fundamental open problem. Although the research frontier has mostly shifted toward ultra-high pressure phases above 400 GPa, we show that even the broken symmetry phase observed around 130~GPa requires revisiting due to its intricate coupling of electronic and nuclear degrees of freedom. Here, we develop a first principle quantum Monte Carlo framework based on a deep neural network wave function that treats both electrons and nuclei quantum mechanically within the constant pressure ensemble. Our calculations reveal an unreported ground-state structure candidate for the broken symmetry phase with $Cmcm$ space group symmetry, and we test its stability up to 96 atoms. The predicted structure quantitatively matches the experimental equation of state and gives the closest x-ray diffraction peak-position match among the tested candidates. Furthermore, our group-theoretical analysis provides a symmetry-counting compatibility check between the $Cmcm$ structure and existing Raman and infrared spectroscopic data. Crucially, static density functional theory calculation reveals the $Cmcm$ structure as a dynamically unstable saddle point on the Born-Oppenheimer potential energy surface, demonstrating that a full quantum many-body treatment of the problem is necessary. These results shed new light on the phase diagram of high-pressure hydrogen and call for further experimental verifications.

Figures & tables

Figure 1: The computational framework. The nuclear coordinates $\mathbf{R}$ and the electron coordinates $\mathbf{r}$ are fed into two neural networks that generate the nuclear wave function $\chi(\mathbf{R})$ and the electronic wave function $\varphi(\mathbf{r},\mathbf{R})$ , respectively. Their product, $\Psi(\mathbf{r},\mathbf{R})=\chi(\mathbf{R})\,\varphi(\mathbf{r},\mathbf{R})$ , defines the total trial wave function. Together with the lattice parameters, they determine the enthalpy of the system $G=E+P_{\mathrm{ext}}\Omega$ , where $P_{\mathrm{ext}}$ is the external pressure and $\Omega$ the cell volume. Treating the enthalpy as the objective function, the lattice parameters $L$ are optimized via simulated annealing, whereas the neural-network parameters $\theta$ are optimized by gradient descent.

Variational ansatz	$E/N$ (Ha/atom)	$\sigma^{2}/N$ (Ha 2 /atom)
SJ-LDA (VMC) [ 49 ]	-0.5195(2)
SJ-LDA (DMC) [ 49 ]	-0.52415(5)
BF-PW (VMC) [ 49 ]	-0.52194(5)	0.025(1)
BF-PW (DMC) [ 49 ]	-0.52610(7)
NQS (VMC) [ 66 ]	-0.52854(9)	0.0088(1)
Present work	-0.529674(5)	0.01575(5)

Table 1: Ground-state energies for

N=54

hydrogen atoms arranged into a BCC lattice with

r_{s}=1.31

treated in full quantum manner. The digits in the brackets represent the statistical uncertainty of the energy and corresponding variance. Details are provided in Sec. IV of the SM [ 1 ] .

Figure 2: The structure with the lowest enthalpy among those tested at 130 GPa for the broken symmetry phase of solid hydrogen. The figure shows the projection of $\mathrm{H}_{2}$ molecules onto the $ab$ plane. Blue and red lines represent hydrogen molecules in the first and second layers along the $c$ axis, respectively. The black dashed line outlines the primitive cell, while the black solid line indicates the conventional cell. 32 hydrogen atoms in the conventional cell fully occupy four $8g$ Wyckoff positions of the space group $Cmcm$ .

Figure 3: Orientation order of the structure shown in Fig. 2 . The angular distribution $g(\theta,\phi)$ illustrates that preferred orientations of $\mathrm{H}_{2}$ molecules are within the $ab$ plane. The color bar denotes the normalized density.

Figure 4: The simulated XRD patterns for various candidate structures at 130 GPa, uniformly rescaled to the experimental volume, compared with the room-temperature observation of Ref. [ 51 ] ; the NNVMC calculation targets

T=0

. For

Cmcm

, both the static profile and a coherent quantum-density-corrected profile obtained from

|\Psi|^{2}

nuclear samples are shown. Details are provided in the SM [ 1 ] . To our knowledge, original XRD profiles (beyond reported peak positions) are unavailable in related literature, with the exception of Ref. [ 52 ] , which focuses on pressures above 200 GPa. This comparison highlights a gap that calls for further experimental validation.

Space group	IR vibron	Raman vibron	IR phonon	Raman phonon
$Pca2_{1}$ [ 23 ]	$A_{1}+B_{1}+B_{2}$	$A_{1}+A_{2}+B_{1}+B_{2}$	$2A_{1}+2B_{1}+2B_{2}$	$2A_{1}+3A_{2}+2B_{1}+2B_{2}$
$P2_{1}/c$ [ 23 ]	$A_{u}+B_{u}$	$A_{g}+B_{g}$	$2A_{u}+B_{u}$	$3A_{g}+3B_{g}$
$P6_{3}/m$ [ 23 ]	$E_{1u}$	$2A_{g}+E_{2g}$	$A_{u}+2E_{1u}$	$2A_{g}+E_{1g}+3E_{2g}$
$Cmcm$	$3B_{2u}+B_{3u}$	$3A_{g}+B_{1g}$	$3B_{2u}+6B_{1u}+3B_{3u}$	$4B_{1g}+7B_{3g}+4A_{g}+5B_{2g}$
Experiments [ 77 , 45 , 43 ]	$2$	$1$	?	?

Table 2: Summary of theoretical predictions and experimental observations on IR and Raman modes. The comparison is based on symmetry-allowed mode counting and the observed features may form a subset of the allowed modes. Question marks (?) denote that the corresponding phonon modes are currently unreported or unresolvable. To our knowledge, the IR phonon is not reported in the literature, and the Raman phonon is reported as a single peak in [ 43 ] , while Hemley et al. [45] observed that this phonon mode becomes extremely weak.

Hyperparameter	Value	Hyperparameter	Value
Dimension of one-electron layer	256	Dimension of two-electron layer	32
Dimension of nuclear layer	32	Number of layers	3
Number of determinants	16	Optimizer	KFAC
Learning rate	3e-2	Learning rate decay	1
Learning rate delay	1e4	Damping	1e-3
Momentum of optimizer	0.0	Batch size	1024

Table S1: Hyperparameters for BCC Train

Hyperparameter	Value	Hyperparameter	Value
Dimension of one-electron layer	256	Dimension of two-electron layer	32
Dimension of nuclear layer	32	Number of layers	3
Number of determinants	8	Optimizer	KFAC
Learning rate	3e-3	Learning rate decay	1
Learning rate delay	1e4	Damping	1e-3
Momentum of optimizer	0.0	Batch size	1024

Table S2: Hyperparameters for Broken Symmetry Phase

Figure S1: Comparison of different annealing strategies. The high-noise strategy (single estimate, more steps) demonstrates faster convergence and lower variance compared to the low-noise strategy (averaged estimates), despite identical computational costs. The shadow in the figure represents the original data points, while the solid line is the smoothed curve via a moving average with a window size of 1000 steps.

Panel of Figure 10 — Figure S2: Left : Training curves for the BCC dynamic lattice at $r_{s}=1.31$ in the NVT ensemble. Right : Training curves for various initial structures at $P_{\text{ext}}=130$ GPa in the NPT ensemble, highlighting the stability of the $Cmcm$ structure. The legend denotes the initial structure. The shadow in the figure represents the original data points, while the solid line is the smoothed curve via a moving average with a window size of 1000 steps.

Figure S3: Training curves for various initial structures at $P_{\text{ext}}=130$ GPa in the NPT ensemble with $2\times 2\times 2$ twist averaged boundary conditions. The legend denotes the initial structure. Shaded regions represent the original data points, while solid lines show smoothed curves obtained via a moving average with a window size of 1000 steps.

Figure S4: The evolution of lattice parameters and volume per atoms as a function of training steps. The legend denotes the initial structure.

Figure S5: Comparison between two NPT training runs starting from a $Cmcm$ structure with 64 atoms at 130 GPa: one with cell angles constrained to $\alpha=\beta=\gamma=90^{\circ}$ (’Fixed angles’) and another with angles freely optimized along with other lattice parameters (’Free angles’). Shaded regions in enthalpy curves represent the original data points, while solid lines show smoothed curves obtained via a moving average with a window size of 1000 steps.

Initial Structure	Final Enthalpy (Ha/atom)
	PBC	TABC
$Cmcm$	-0.483666(10)	-0.477739(6)
$Pca2_{1}$	-0.482817(9)	-0.475292(6)
$P2_{1}c$	-0.482795(9)	-0.475416(6)
$P6_{3}/mmc$ with aligned molecule	-0.47791(1)	-0.45908(1)
$P6_{3}m$	-0.46987(2)	-0.45477(2)

Table S3: Final enthalpies based on inference with 10000 steps after training, including PBC and TABC.

Space group	Lattice parameters (Bohr, deg)	Wyckoff position	Fractional coordinates
Space group	Lattice parameters (Bohr, deg)	Wyckoff position	$x$	$y$	$z$
$Cmcm$	$a=8.48$	8g	0.5992	0.3967	0.2500
	$b=9.60$	8g	-0.0825	0.2402	0.2500
	$c=5.47$	8g	0.7642	0.3967	0.2500
	$\alpha=\beta=\gamma=90.0$	8g	0.4175	0.0867	0.2500

Table S4: The structure identified at 130 GPa for the broken symmetry phase.

Candidate	$N_{\rm all}$	$\Delta_{\rm sim\to exp}$	$\Delta_{\rm exp\to sim}$	$N_{\rm near}$	$\Delta_{\rm sim\to exp}$	$\Delta_{\rm exp\to sim}$
	All selected peaks			Near experimental peaks
$Cmcm$	7	0.1248	0.0036	4	0.0039	0.0036
$P2_{1}/c\text{-}8$	9	0.1482	0.0116	6	0.0178	0.0116
$P2_{1}/c\text{-}24$	12	0.1447	0.0096	7	0.0286	0.0096
$Pca2_{1}$	8	0.1570	0.0078	5	0.0089	0.0078
$P6_{3}/m$	8	0.1603	0.0078	4	0.0220	0.0078

Table S5: RMS peak-position comparison after uniform volume rescaling. The left block includes all simulated peaks above the common intensity threshold; the right block is restricted to

1.35\leq d\leq 1.65

Å. All RMS differences are in Å.

Figure S6: The structures used in the XRD patterns in Fig. 4 of the main text. The solid line represents the bonds between the hydrogen atoms, while the box represents the conventional cell for $Cmcm$ and primitive cell for other structures.

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