High-pressure hydrogen phase diagram from quantum Monte Carlo and machine learning

High-pressure hydrogen is of paramount importance in several fields, including planetary science, condensed matter physics, and energy production applications. Despite its significance, many properties of this system are still not fully understood, due to the difficulty of realizing the required extreme conditions in the laboratory and probing the compressed samples. Numerical results are thus extremely valuable. Quantum Monte Carlo (QMC) algorithms have been proven to be among the most effective methods for describing the physics and properties of high-pressure hydrogen, although their large computational cost limits their applicability to small systems. In this thesis, we discuss techniques that aim at combining the accuracy of QMC methods with the efficiency of machine learning potentials (MLPs). In particular, we employ the ∆-learning framework together with kernel ridge regression, and train models on the difference between QMC reference calculations and a computationally cheaper "baseline potential", which in our case was obtained with the density functional theory (DFT) method. This approach allows us to reach a higher accuracy with relatively small datasets, a crucial feature for resource-heavy algorithms like QMC. We also analyze the bias affecting both forces and pressures within the variational Monte Carlo (VMC) method when the wave function employed is not fully optimized, and propose a suitable correction. We present two applications of our framework to high-pressure hydrogen. In the first one, we determined the deuterium Hugoniot with MLPs trained on both variational and diffusion Monte Carlo. We find a good agreement with experiments, even though our results suggest a slightly more compressible system for large pressures. In the second application, we study the hydrogen liquid-liquid phase transition (LLPT). We discuss results obtained with two MLPs trained on VMC and DFT data, respectively. For the latter, we employed MACE, a message passing neural network, to study the order of the transition in the thermodynamic limit. Our results predict a first-order transition between a defective molecular solid and an atomic liquid close to the melting line, and a liquid-liquid crossover at higher temperatures.