Compute the large-wave-vector static exchange local-field factor in two dimensions

Compute the static first-order exchange local-field factor G_x(q) of the strictly two-dimensional electron gas in the large-wave-vector limit q/k_F\to\infty.

Background

The paper compares the exact first-order exchange local-field factor G_x(q) with the Hubbard approximation over the range 0<q<2k_F and discusses the distinct static and dynamic large-q limits. It notes that the Hubbard approximation approaches the dynamic high-frequency exchange-only value, whereas the static first-order limit is a different quantity.

The static large-q limit is known in three dimensions but had not been calculated in the two-dimensional system treated in the paper. Determining this limit would complete the characterization of the exchange-only static local-field factor beyond the 2k_F region.

References

The static first-order $G_x(q\to\infty)$ is a different limit (it is $1/3$ in 3D), which we have not computed in 2D.

— Exchange self-energy and vertex corrections to static screening in the two-dimensional electron gas  (2610.08309 - Sarma, 6 Oct 2026) in Section 'Comparison with the Hubbard local-field factor,' subsection 'Intermediate and large wave vectors'