A deep variational free energy approach to dense hydrogen

We present a deep generative model-based variational free energy approach to the equations of state of dense hydrogen. We employ a normalizing flow network to model the proton Boltzmann distribution and a fermionic neural network to model the electron wavefunction at given proton positions. By jointly optimizing the two neural networks we reached a comparable variational free energy to the previous coupled electron-ion Monte Carlo calculation. Our result suggests that hydrogen in the planetary condition is even denser compared to previous Monte Carlo and ab initio molecular dynamics data, which is further away from the empirical chemical model predictions. Obtaining reliable equations of state of dense hydrogen, and in particular, direct access to entropy and free energy opens new opportunities in planetary modeling and high-pressure physics research.