Determine the asymptotic fate of the exchange-induced Friedel logarithm

Determine whether higher-order interaction corrections exponentiate the first-order logarithm in the two-dimensional electron gas, thereby modifying the long-distance Friedel-oscillation decay law, or instead cut off the logarithm and leave only an amplitude renormalization.

Background

The first-order exchange correction to the static polarizability develops a two-sided singularity at q=2k_F. In real space, this produces a Friedel-oscillation contribution whose amplitude grows approximately as ln(k_F r), causing first-order perturbation theory to fail asymptotically when r_s ln(k_F r) becomes of order unity.

The paper identifies two unresolved possibilities for higher-order corrections: they may resum or exponentiate the logarithmic enhancement, changing the decay law, or they may regulate it so that the oscillations retain the usual decay form with only a renormalized amplitude. The first-order calculation cannot distinguish between these scenarios.

References

Whether higher orders exponentiate the logarithm, modifying the decay law as interactions do in one dimension, or cut it off, leaving an amplitude renormalization as assumed in local-field theory, cannot be decided at first order.

— Exchange self-energy and vertex corrections to static screening in the two-dimensional electron gas  (2610.08309 - Sarma, 6 Oct 2026) in Section 'Friedel oscillations: RPA versus RPA + exchange,' subsection 'Interpretation'