We have performed quantum Monte Carlo (QMC) [12, 13] calculations for the 2D fluid and Wigner crystal phases, achieving much smaller statistical error bars than previous studies [6, 7, 14]. We used the variational and diffusion quantum Monte Carlo (VMC and DMC) methods as implemented in the casino code [15]. DMC is the most accurate method available for studying quantum many-body systems such as electron gases. Fermionic symmetry is imposed via the fixed-node approximation [16], in which the nodal surface is constrained to equal that of a trial wave function. The Ewald method was used to calculate the electron-electron interaction energy [17]. We used extremely flexible Slater-Jastrow-backflow [18] wave functions, and we optimized the crystal orbitals by directly minimizing the DMC energy. Our fixed-node DMC energies are therefore more accurate than those of earlier calculations. Finally, we have dealt with finite-size effects in a more sophisticated fashion than previous studies [19]. 

It is well-established that the triangular crystal has a considerably lower energy than competing lattices in the density range of interest. A triangular lattice was therefore used in all our crystal calculations. Antiferromagnetism is frustrated on a triangular lattice, and antiferromagnetic crystals are expected to form a spin liquid [20]. Instead, we have considered antiferromagnetic crystals consisting of alternating lines of spin-up and spin-down electrons. 

We used the Jastrow factor proposed in Ref. [21]. Plane-wave orbitals exp( i G · r ) were used in the Slater wave function in our fluid calculations, while identical Gaussian orbitals exp( − Cr $^{2}$) centered on lattice sites were used in our crystal calculations. The Slater wave function was evaluated at quasiparticle coordinates defined by the backflow (BF) transformation described in Ref. [18]. The free parameters in the Jastrow factor and BF function were optimized within VMC by minimizing the energy [22]. The BF functions for antiparallel spins are much larger than those for parallel spins, which are already kept apart by Pauli exclusion. BF is less important in the crystal because the localization of the orbitals keeps the electrons apart. For example, at r$_{s}$ = 30 a.u. [23], BF lowers the DMC energy of the 42-electron paramagnetic fluid by 0.000036(3) a.u. per electron, whereas it lowers the energy of the 64-electron ferromagnetic Wigner crystal by only 0.0000023(3) a.u. per electron. Time-step bias was removed from our final DMC energies by linear extrapolation to zero time step [24]. A target population of at least 1500 configurations was used in our production runs, making population-control bias negligible [24]. 

We optimized the Gaussian exponent C in the crystal orbitals by minimizing the Slater-Jastrow DMC energy. At r$_{s}$ = 40 a.u., the DMC-optimized exponent of the ferromagnetic crystal ( C F DMC ≈ 0 . 0003 a.u.) is smaller than the exponent obtained by minimizing the VMC energy ( C F VMC ≈ 0 . 0006 a.u. with our Jastrow factor), which is, in turn, substantially smaller than the exponent within either Hartree or Hartree-Fock theory ( C F H = 0 . 0019 a.u.) 

