Message-passing neural quantum states for the homogeneous electron gas

We introduce a neural-network-based wave function Ansatz to accurately simulate extended, strongly interacting fermions in continuous space. The variational state is parameterized via permutation-equivariant message-passing neural networks to transform single-particle coordinates into highly correlated quasi-particle coordinates. We show the versatility and accuracy of this Ansatz by simulating the ground state of the homogeneous electron gas in three spatial dimensions at different densities and system sizes. Our model respects the fundamental symmetries of the Hamiltonian, such as continuous translation symmetries. We show better or comparable ground-state energies on small benchmark systems, compared to current state-of-the-art neural-network wave functions, using orders of magnitudes less variational parameters and optimization steps. These savings allow us to scale up to system sizes of $N=128$ electrons, previously inaccessible to neural-network wave functions in continuous space, opening the door for future work on finite-size extrapolations to the thermodynamic limit. We investigate the Ansatz's capability of identifying and representing different phases of matter without imposing any structural bias toward a given phase.