Deep Variational Free Energy Calculation of Hydrogen Hugoniot

TL;DR

This work introduces a deep variational free-energy framework to compute the equation of state and Hugoniot of hydrogen in the warm dense matter regime by jointly optimizing three neural models: a normalizing flow for nuclear Boltzmann sampling, a masked autoregressive network for finite-temperature electronic excitations, and a quantum flow-based transformation of Hartree-Fock states to form the electronic wavefunctions. The approach enables finite-temperature, excited-state electronic effects to be incorporated into a variational density-matrix, circumventing the fermion sign problem and connecting finite-temperature methods with ground-state calculations. Results for deuterium across multiple system sizes, densities, and temperatures show sensible agreement with experimental data and other theories, while providing insight into electron occupation, RDFs, and the role of temperature in dissociation and metallization processes. The method offers a robust benchmark for EOS in the WDM region and a path toward consistent cross-regime predictions crucial for planetary modeling and inertial confinement fusion.

Abstract

We develop a deep variational free energy framework to compute the equation of state of hydrogen in the warm dense matter region. This method parameterizes the variational density matrix of hydrogen nuclei and electrons at finite temperature using three deep generative models: a normalizing flow model for the Boltzmann distribution of the classical nuclei, an autoregressive transformer for the distribution of electrons in excited states, and a permutational equivariant flow model for the unitary backflow transformation of electron coordinates in Hartree-Fock states. By jointly optimizing the three neural networks to minimize the variational free energy, we obtain the equation of state and related thermodynamic properties of dense hydrogen for the temperature range where electrons occupy excited states. We compare our results with other theoretical and experimental results on the deuterium Hugoniot curve, aiming to resolve existing discrepancies. Our results bridge the gap between the results obtained by path-integral Monte Carlo calculations at high temperature and ground-state electronic methods at low temperature, thus providing a valuable benchmark for hydrogen in the warm dense matter region.

Figures (4)

• Figure 1: Computational graph for the finite electron temperature variational free energy calculation. The model consists of three trainable components: a normalizing flow Eq. (\ref{['eq:boltzmann']}) for the nucleus Boltzmann distribution $p(\boldsymbol{s})$ in the blue part; a variational autoregressive network Eq. (\ref{['eq:transformer1']}), Eq. (\ref{['eq:transformer2a']}), and Eq. (\ref{['eq:transformer2b']}) for the electron excitation distribution $p(\boldsymbol{k}|\boldsymbol{s})$ in the red part; and a quantum flow for the electron wave functions $\Psi_{\boldsymbol{s},\boldsymbol{k}}(\boldsymbol{r})$ shown in Eq. (\ref{['eq:wavefunction']}) in the purple part. We jointly optimize the three neural networks to minimize the variational free energy $F$ in Eq. (\ref{['eq:FreeEnergy']}).
• Figure 2: Training curve of variational free energy method for $N=32$, $T=31\,250$ K, $r_s=2$ deuterium system, with a 10-epoch moving average. The red line in the main figure shows the variational free energy $F$ per atom and the error bar (red-filled area) during training. The red lines in the set insets show the internal energy $E$ per atom and the pressure $P$ change. The brown lines are the PBC $N=32$ results from RPIMC PhysRevLett.85.1890, and the filled areas denote the error bars.
• Figure 3: Electron distribution on single-particle orbitals and nucleus-nucleus RDF of $N=32$, $r_s=2$ deuterium system at several temperatures. (a) Electron occupation of single-particle orbitals. $E_i$ are Hartree-Fock energy levels of these orbitals. The dashed black line is ground state occupation as reference. (b) Nucleus-nucleus RDF.
• Figure 4: Pressure-compression ratio diagram of Hugoniot experiments and computations. The red symbols are Lagrange interpolation of variational free energy results of $N=32$ system, the yellow star is the result at $10\,000$ K for the $N=54$ system. The experimental and theoretical results are from Z-machine PhysRevLett.118.035501, laser PhysRevLett.122.255702, DFT molecular dynamics (DFT) PhysRevB.83.094101, DFT with finite temperature XC functionals (FT-DFT) PhysRevLett.118.035501, direct PIMC filinov_calculation_2005, RPIMC PhysRevLett.85.1890, and CEIMC PhysRevLett.115.045301PhysRevB.102.144108.