4 DMC energy (10 $_{−}$2 a.u. / elec.) $^{r}$s Ferro. crystal Antif. crys. Para. fluid Ferro. fluid 15 − 5 . 966 5(1) · · · · · · · · · 20 − 4 . 619 5(2) − 4 . 622 9(2) − 4 . 630 5(4) − 4 . 621 3(3) 25 − 3 . 773 1(2) − 3 . 775 1(3) − 3 . 777 4(2) − 3 . 774 0(2) 30 − 3 . 191 7(2) − 3 . 192 2(2) − 3 . 192 6(1) − 3 . 191 3(1) 35 − 2 . 766 9(1) − 2 . 767 2(1) − 2 . 766 5(1) − 2 . 765 7(1) 40 − 2 . 443 2(1) − 2 . 443 1(2) − 2 . 441 6(1) − 2 . 441 6(1) 45 − 2 . 188 1(1) − 2 . 187 5(2) · · · · · · 50 − 1 . 981 7(2) − 1 . 981 4(2) · · · · · · TABLE I: DMC energy as a function of $^{r}$s for ferromagnetic and antiferromagnetic triangular Wigner crystals, and paramagnetic and fully ferromagnetic Fermi fluids. All results have been extrapolated to zero time step and infinite system size. fluids at r$_{s}$ = 20 a.u. than Attaccalite et al. [14], resulting in somewhat smaller values of the spin susceptibility. The ground-state energies of the different phases of the 2D HEG are plotted against r$_{s}$ in Fig. 1. Unlike previous QMC studies, our statistical error bars are sufficiently small that we can resolve the energy difference between the ferromagnetic and paramagnetic fluids. We can also identify the crystallization density with much greater precision. Previous studies found crystallization to occur at r$_{s}$ = 37(5) a.u. [6] and r$_{s}$ = 34(4) a.u. [7]. We find the crystallization density to be r$_{s}$ = 31(1) a.u., but the transition is from the paramagnetic fluid to an antiferromagnetic crystal, not the ferromagnetic fluid to the ferromagnetic crystal as found in the previous studies. Our calculations locate the density at which the paramagnetic fluid becomes unstable to the ferromagnetic fluid at about r$_{s}$ = 40 a.u., but at this density the Wigner crystal is more stable than either fluid phase, so we do not find a region of stability for the ferromagnetic fluid. We have looked for a region of stability for a partially spin-polarized fluid by calculating the energy of a fluid of spin polarization ζ = 2 / 5 at r$_{s}$ = 35 a.u. The DMC energy, extrapolated to zero time step and infinite size, is − 0 . 027666(1) a.u. per electron, which is not significantly less than the energies of the paramagnetic and ferromagnetic fluids, but is significantly higher than the crystal energy (see Table I). It is therefore unlikely that a region of stability exists for a partially spin-polarized fluid. We do, however, find a transition from an antiferromagnetic Wigner crystal to a ferromagnetic one at r$_{s}$ = 38(5) a.u. This is considerably higher than the density at which Bernu et al. [20] found a transition from a spin liquid to a ferromagnetic crystal using a multispin exchange model ( r$_{s}$ = 175 a.u.).