Missing derivative discontinuity of the exchange-correlation energy for attractive interactions: The charge Kondo effect

Abstract

We show that the energy functional of ensemble density functional theory (DFT) [Perdew et al., Phys. Rev. Lett. 49, 1691 (1982)] in systems with attractive interactions is a convex function of the fractional particle number $N$ and is given by a series of straight lines joining a subset of ground-state energies. As a consequence the exchange-correlation (XC) potential is not discontinuous for all $N$. We highlight the importance of this exact result in the ensemble-DFT description of the negative-$U$ Anderson model. In the atomic limit the discontinuity of the XC potential is missing for odd $N$ while for finite hybridizations the discontinuity at even $N$ is broadened. We demonstrate that the inclusion of these properties in any approximate XC potential is crucial to reproduce the characteristic signatures of the charge-Kondo effect in the conductance and charge susceptibility.