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

TL;DR

This work introduces a beyond-Born–Oppenheimer neural-network variational Monte Carlo (NNVMC) framework to solve the full electron–nuclear quantum problem under constant pressure for solid hydrogen. By representing both nuclear and electronic wavefunctions with flexible neural networks and performing enthalpy minimization, the authors identify an orthorhombic Cmcm ground-state candidate at 130 GPa that aligns with XRD and vibrational spectroscopy data, while static BO-DFT finds Cmcm to be a saddle point. They demonstrate that strong nuclear quantum effects and non-adiabatic couplings stabilize Cmcm, a result inaccessible to conventional BO approaches. The study provides a comprehensive validation against experiment, clarifies the role of NQEs in the hydrogen phase diagram, and introduces a general methodology for predicting quantum materials under finite pressure.

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 X-ray diffraction patterns. Furthermore, our group-theoretical analysis shows that the $Cmcm$ structure is compatible with 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 (9)

• 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.
• Figure 2: The structure with lowest enthalpy discovered at 130 GPa for the broken symmetry phase 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 $c$ 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. \ref{['fig:structure']}. 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 pattern for various candidate structures at 130 GPa compared with the experimental observation ji_ultrahighpressureisostructural_2019.
• 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.