Metal-insulator transition in the Hartree-Fock phase diagram of the fully polarized homogeneous electron gas in two dimensions

We determine numerically the ground state of the two-dimensional fully polarized electron gas within the Hartree-Fock approximation without imposing any particular symmetries on the solutions. At low electronic densities, the Wigner crystal solution is stable, but for higher densities ($r_s$ less than $\sim 2.7$) we obtain a ground state of different symmetry: the charge density forms a triangular lattice with about 11% more sites than electrons. We prove analytically that this conducting state with broken translational symmetry has lower energy than the uniform Fermi-gas state in the high-density region giving rise to a metal to insulator transition.

Figure 1

(Color online) Extrapolated energies $E_{\infty }-E_{\text{FG}}$ (milli-Hartree units) versus $r_s$. Points with error bars: present calculations, full line (red): data of Ref. 8, full line (green): fit to present results for $\left(r_s\lesssim 2.7\right)$, vertical dash line (blue): $r_s\sim 2.7$.

Figure 2

(Color online) Left: charge density $\rho \left(\mathbf{r}\right)/⟨\rho ⟩-1$, with $\rho \left(\mathbf{r}\right)={\sum }_{i=1}^N{\left|{\phi }_i\left(\mathbf{r}\right)\right|}^2$. Right: $\tilde{\rho }\left(\mathbf{k}\right)$, the Fourier transform of the charge density, where $\tilde{\rho }\left(0\right)=N$ has been removed. The grid points are those compatible with the periodic conditions. Top: the number of maxima is $N_c=9^2+5^2+9\times 5=151$. Bottom: the number of maxima is $N_c={12}^2+1^2+12\times 1=157$. Gray levels corresponds to the same density in both figures. Colored lines correspond to ${\mathbf{L}}_1=l{\mathbf{e}}_1+m{\mathbf{e}}_2$, where the numbers stand for $l$, $m$ (see text).

Figure 3

(Color online) Numerical values of $b_k$ for 499 electrons in two dimensions at $r_s=2$.

Figure 4

(Color online) Effect of the screening on the ground-state phase diagram for $N=61$, 151, and 499. The different phases are labeled “FG” (Fermi-gas ground state), “M” (metallic ground state), and “W” (Wigner crystal). For all values of the screening parameter $\alpha$ of the screened Coulomb potential $V\left(q\right)$, we have found a metallic phase of energy lower than the Fermi gas and the Wigner crystal energy.

Figure 5

The circle is the Fermi surface. The shaded surfaces are the regions where $b\left(k\right)$ is nonzero. The new state ${\psi }_k$ mixing ${\varphi }_k$ and ${\varphi }_{k+Q_k}$ is now resonant with ${\psi }_{k^{\prime }}$.