A representability problem in density functional theory for superconductors

TL;DR

This paper investigates the foundational issue of noninteracting $v$-representability in density functional theory for superconductors, revealing that such a Kohn-Sham system does not exist at zero temperature, yet the theory remains practically useful for weakly correlated systems.

Contribution

The paper proves the nonexistence of a noninteracting Kohn-Sham system at zero temperature in superconducting DFT, challenging a key assumption of the theory.

Findings

Noninteracting $v$-representability fails at zero temperature.

Density functional theory remains effective for weakly correlated superconductors.

Theoretical foundation of superconducting DFT requires reconsideration.

Abstract

Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn-Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system. However, the question of noninteracting $v$-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist in at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.