  We present a study of spin-unpolarized and spin-polarized two-dimensional
uniform electron liquids using variational and diffusion quantum Monte Carlo
(VMC and DMC) methods with Slater-Jastrow-backflow trial wave functions.
Ground-state VMC and DMC energies are obtained in the density range $1 \leq
r_\text{s} \leq 40$. Single-particle and many-body finite-size errors are
corrected using canonical-ensemble twist-averaged boundary conditions and
extrapolation of twist-averaged energies to the thermodynamic limit of infinite
system size. System-size-dependent errors in Slater-Jastrow-backflow DMC
energies caused by partially converged VMC energy minimization calculations are
discussed. We find that, for $1 \leq r_\text{s} \leq 5$, optimizing the
backflow function at each twist lowers the twist-averaged DMC energy at finite
system size. However, nonsystematic system-size-dependent effects remain in the
DMC energies, which can be partially removed by extrapolation from multiple
finite system sizes to infinite system size. We attribute these nonsystematic
effects to the close competition between fluid and defected crystal phases at
different system sizes at low density. The DMC energies in the thermodynamic
limit are used to parameterize a local spin density approximation correlation
functional for inhomogeneous electron systems. Our zero-temperature phase
diagram shows a single transition from a paramagnetic fluid to a hexagonal
Wigner crystal at $r_\text{s}=35(1)$, with no region of stability for a
ferromagnetic fluid.
