Figure 1

Correlation energy per particle in atomic units for deltium, the one-dimensional uniform contact-interacting $(\lambda =1)$ Fermi gas, in the high-density limit. The solid line is the exact result calculated from the solutions of the Bethe-Ansatz integral equations. The long-dashed line is the simple local-density approximation (LDA) given by Eq. (20) in the original reference.Reuse & Permissions

Figure 2

The same as Fig. 1 but for the low-density limit.Reuse & Permissions

Figure 3

Correlation energy in atomic units for diracium, the one-dimensional analog of helium $(\lambda =1)$. The solid line is the exact result, the short-dashed line is the LDA result, and the long-dashed line is the second-order perturbation theory result. Note that we extract the correlation energy approximately by subtracting the self-consistent exact-exchange total energy from the exact total energy. The exact-exchange density becomes an unreliable approximation to the exact density as $Z$ decreases and at $Z=1∕2$ the density is even qualitatively wrong as the self-consistent exact-exchange solution is no longer bound. To avoid this complication, we terminate the plot at $Z=1$, where we still expect the self-consistent exact-exchange density to be an accurate representation of the exact density.Reuse & Permissions

Figure 4

The behavior of the density in atomic units for diracium $(Z=1)$ at various interaction strengths, $\lambda$. We plot $d\ln n\left(x\right)∕dx$ to highlight the asymptotic behavior of the density. For $\lambda >{\lambda }_{\mathit{crit}}$, the system is ionized.Reuse & Permissions

Figure 5

Comparison of the self-consistent LDA density in atomic units (dashed line) with the exact (solid line) for $\lambda =1$ and $Z=0.5$. This is an extreme case where the exact-exchange formalism no longer binds.Reuse & Permissions

Figure 6

The expectation value of the interaction in atomic units at various interaction strengths, $\lambda$. Beyond $\lambda ={\lambda }_{\mathit{crit}}\left(2.6517\right)$, the system is ionized and the interaction energy vanishes.Reuse & Permissions