Hartree-Fock Ground State of the Three-Dimensional Electron Gas

In 1962, Overhauser showed that within Hartree-Fock (HF) the electron gas is unstable to a spin-density wave state. Determining the true HF ground state has remained a challenge. Using numerical calculations for finite systems and analytic techniques, we study the unrestricted HF ground state of the three-dimensional electron gas. At high density, we find broken spin symmetry states with a nearly constant charge density. Unlike previously discussed spin wave states, the observed wave vector of the spin-density wave is smaller than $2k_F$. The broken-symmetry state originates from pairing instabilities at the Fermi surface, a model for which is proposed.

Figure 1

Energy differences (in Ry) between the UHF ground state and the RHF state. (a) The energy lowering per electron, $\delta E=(E-E_{\mathrm{RHF}})/N$, vs $r_s$ for different values of N [12]. (b) The three components of $\delta E$, kinetic ($K$), Hartree ($V_H$), and exchange ($V_{\mathrm{ex}}$) for $N=54$ (solid lines) and $N=66$ (dashed lines). The inset is an enlargement of the low $r_s$. Error bars are estimated from the results for various ${\mathbf{k}}_{\theta }$ points.

Figure 2

Momentum distribution $n(\mathbf{k})$ at different $r_s$ vs $k/k_F$. The main graph has $N=54$, with $\mathit{\theta }=(-0.368,0.172,-0.364)$. The arrow indicates $k_F$. The inset shows $[n(\mathbf{k})-n_{\mathrm{RHF}}(\mathbf{k})]$ at $r_s=1$ for $N=246$, with $\mathit{\theta }=(-0.494,-0.425,0.144)$. Note the primary spike at $k_F$, and the paired smaller spikes with one on each side of $k_F$.

Figure 3

Left: Peak values and locations of the Fourier transforms of spin and charge densities for different values of $r_s$ in $N=54$. Errors are estimated from the values at different ${\mathbf{k}}_{\theta }$ points. Note the logarithmic scale in $S(q)$. Right: Contour plot of the spin density $\sigma (x,y,z)$ for a slice parallel to the $x\mathrm{\text{−}}y$ plane. The system has $N=512$ and $r_s=1$. The $\mathbf{q}$-vector $[2\overline{6}\overline{3}]$ has the leading $S_{\sigma }(\mathbf{q})$ value, followed by $[33\overline{3}]$, and then $[4\overline{1}5]$ with $S_{\sigma }$ 4 times smaller.

Figure 4

Cartoon of pairing state with $q_{\sigma }<2k_F$. The primary pairing state $\{\mathbf{k},{\mathbf{k}}^{\prime }\}$ is at the FS. The satellite pairing state $\{\mathbf{s},{\mathbf{s}}^{\prime }\}$ can be in the shaded areas, within $\mathcal{O}(2\pi /L)$ of $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$. Dashed arrow lines illustrate Overhauser [3, 7] pairing at $2k_F$.