Broadening of the derivative discontinuity in density functional theory

Abstract

We clarify an important aspect of density functional theories, the broadening of the derivative discontinuity (DD) in a quantum system, with fluctuating particle number. Our focus is on a correlated model system, the single level quantum dot in the regime of the Coulomb blockade. We find that the DD-broadening is controlled by the small parameter Γ/U , where Γ is the level broadening due to contacting and U is a measure of the charging energy. Our analysis suggests, that Kondoesque fluctuations have a tendency to increase the DD-broadening, in our model by a factor of two.PACS numbers: 05.60.Gg, 85.65.+h Introduction. Over the years the density functional theory (DFT) developed into an important tool to study transport properties of nano-systems and single molecules. 1-5 This development occured despite of the fact, that often the results are quantitatively sensitive to the approximations made for the exchange correlation (XC) functional, V XC [n], underlying such calculations. 6-10 In principle, dc-transport calculations should combine either long-time evolution of wavepackets or a KS-based quasistationary formalism, in both cases with special dynamical XC-functionals. 9,11,12 In practice, the available ground state functionals are being used.The neglect of dynamical correlations for simulation of dc-transport was justfied for spinless systems for which a Friedel-sum rule holds. 13 In such systems, approximations to the XC-potential of the ground state introduce the largest numerical error in the regime of Coulomb blockade (CB) where the system ("quantum dot", QD) is only weakly coupled to the electronic reservoirs and the filling is close to an integer. Its signature is an addition energy, U , that largely exceeds the (single particle) level spacing of the QD.In this Letter we exploit the observation that CB is intrinsically an equilibrium phenomenon even though it is mostly discussed in its effect on transport measurements 15 ; in a broader context it is a typical manifestation of the reduced compressibility, dn(r)/dµ, of repulsively interacting fermion gases. Therefore, it has a reincarnation in XC-functionals of DFT where it appears as the derivative discontinuity (DD).Starting from the seminal work by Perdew et al. 16 , the DD was almost exclusively discussed in the limiting case of decoupled quantum dots, i.e. closed systems. There, the XC-functional jumps discontinuously when tuning the particle number, N , of a closed system in its ground state through an integer value

Interacting fermions in one-dimensional disordered lattices: Exploring localization and transport properties with lattice density-functional theories

“…74, using support from small Anderson clusters, and fully discussed in Refs. 58,[75][76][77]. According to these considerations, and also for numerical convenience, the XC potential was slightly smoothed in our actual calculations.…”

“…58,[75][76][77]105,112,113,[117][118][119] These studies show that for single-channel molecular junctions, and at zero temperature, the exactness of the KS conductance is ensured by the Friedel's sum rule. At the same time, the same works (see especially Refs.…”

Lattice density functional theory at finite temperature with strongly density-dependent exchange-correlation potentials

“…Let us consider one single-orbital site in contact with a heat and particle bath at inverse temperature β and chemical potential µ 22,23 .…”

Solving lattice density functionals close to the Mott regime

“…[37]) among which LDFT has become particularly useful. 21,38,39 Static and time-dependent [14][15][16] lattice DFT were indeed used to investigate the physics of Hubbard models with on-site interaction 13,[17][18][19][20][21][22] , Kondo models [23][24][25][26][27][28] , disordered interacting lattice models 31,32 and spin liquids 33 .…”

“…Our results are therefore relevant for all 1D lattice models where BALDA is applied, including Kondo systems 23,[25][26][27][28] , dynamical Coulomb blockade treated with time-dependent DFT 24 , harmonically trapped Hubbard electrons 19,22,33 and spinless Fermions with neighboring interaction 18 . µ-BALDA is actually a rather general approach and it could be in principle generalized also to higher dimension or to treat other discontinuous functionals.…”