We present a novel combination of quantum Monte Carlo methods and a finite size extrapolation framework with which we calculate the thermodynamic limit of the exact correlation energy of the polarized electron gas at high densities to meV accuracy, −40.44(5) and −31.70(4) mHa at rs = 0.5 and 1, respectively. The fixed-node error is characterized and found to exceed 1 mHa, and we show that the magnitude of the correlation energy of the polarized electron gas is underestimated by up to 6 meV by the Perdew-Wang parametrization, for which we suggest improvements.

We show how a ground state trial wavefunction of a Fermi liquid can be systematically improved introducing a sequence of renormalized coordinates through an iterative backflow transformation. We apply this scheme to calculate the ground state energy of liquid 3 He in two dimensions at freezing density using variational and fixed-node diffusion Monte Carlo. Comparing with exact transient estimate results for systems with small number of particles, we find that variance extrapolations provide accurate results for the true ground state together with stringent lower bounds. For larger systems these bounds can in turn be used to quantify the systematic bias of fixed-node calculations. These wave functions are size consistent and the scaling of their computational complexity with the number of particles is the same as for standard backflow wave functions.

We show that the recently introduced iterative backflow wave function can be interpreted as a general neural network in continuum space with nonlinear functions in the hidden units. Using this wave function in variational Monte Carlo simulations of liquid ^{4}He in two and three dimensions, we typically find a tenfold increase in accuracy over currently used wave functions. Furthermore, subsequent stages of the iteration procedure define a set of increasingly good wave functions, each with its own variational energy and variance of the local energy: extrapolation to zero variance gives energies in close agreement with the exact values. For two dimensional ^{4}He, we also show that the iterative backflow wave function can describe both the liquid and the solid phase with the same functional form-a feature shared with the shadow wave function, but now joined by much higher accuracy. We also achieve significant progress for liquid ^{3}He in three dimensions, improving previous variational and fixed-node energies.

Quantum Monte Carlo simulations at zero temperature of an ensemble of 3He atoms adsorbed on Mg and Alkali substrates yield strong evidence of a thermodynamically stable liquid 3He monolayer on all Alkali substrates, with the possible exception of Li. The effective two-dimensional density is θ≈0.02  Å-2 on Na, making it the lowest density liquid in nature. Its existence is underlain by zero-point atomic motion perpendicular to the substrate, whose effect is softening the short-range repulsion of the helium interatomic potential. The monolayer films should turn superfluid at a temperature Tc∼1  mK. No liquid film is predicted to form on Mg, or on stronger substrates such as graphite.