Exact ground-state properties of the one-dimensional electron gas at high density

The dynamical response theory is used to obtain an analytical expression for the exchange energy of a quantum wire for arbitrary polarization and width. It reproduces the known form of exchange energy for a one-dimensional (1D) electron gas in the limit of infinitely thin cylindrical and harmonic wires. The structure factor for these wires is also obtained analytically in the high-density or small $r_s$ limit. This structure factor allows to get the exact correlation energy for both the wires and demonstrates that there are at least two methods to get the ideal Coulomb limit in one dimension. The analytical expression for the pair correlation function is also presented for small distances and provides a justification for the small $r_s$ expansion as long as $r_s<\frac{3}{2}\left(\frac{{\pi }^2}{{\pi }^2+3}\right)=1.15$.

Figure 1

VMC static structure factor (SSF) of an infinitely thin wire with $N=99$, compared with the high-density theory (solid line). The main plot shows the SSF for $r_s=0.7$, and the inset is for $r_s=0.9$.

Figure 2

The pair correlation function $g\left(r\right)$ of 1D homogeneous electron gas (27) in infinitely thin wires. The variational Monte Carlo (VMC) simulation data [25] are compared with the high-density expansion (24) at $r_s=0.7$.

Figure 3

The small-$r$ pair correlation function $g\left(r\right)$ for $r_s=0.1–0.5$ from left to right where VMC data (points) are compared to formula (29) (thick lines).

Figure 4

Graphically interchanging the order of integrations from $dtd\overline{x}$ to $d\overline{x}dt$ for $x<1$.