Exact Kohn-Sham eigenstates versus quasiparticles in simple models of strongly correlated electrons

We present analytic expressions for the exact density functional and Kohn-Sham Hamiltonian of simple tight-binding models of correlated electrons. These are the single- and double-site versions of the Anderson, Hubbard, and spinless fermion models. The exact exchange and correlation potentials keep the full nonlocal dependence on electron occupations. The analytic expressions allow us to compare the Kohn-Sham eigenstates of exact density functional theory with the many-body quasiparticle states of these correlated-electron systems. The exact Kohn-Sham spectrum describes correctly many of the nontrivial features of the many-body quasiparticle spectrum such as, for example, the precursors of the Kondo peak. However, we find that some pieces of the quasiparticle spectrum are missing because the many-body phase space for electron and hole excitations is richer.

Figure 1

Three-dimensional plot of the exact energy functional $Q$ of the single-site Anderson-Hubbard model as a function of $(n_{d,\uparrow },n_{d,\downarrow })$, for a value of ${\epsilon }_d=1$ and of $U=10$ (in arbitrary units).

Figure 2

Quasiparticle spectrum of the single-site Anderson-Hubbard model for $M=0$. The shaded gray area shows the position of the many-body poles, where the area is proportional to the weight of the peak. The black solid line represents the location of the exact Kohn-Sham eigenstate. The poles of the paramagnetic mean-field solution are shown with a dashed red line. Energy units are arbitrary.

Figure 3

Quasiparticle spectrum of the single-site Anderson-Hubbard model for a maximally spin-up polarized case. (a) and (b) show the poles of the spin-up and spin-down Green's functions, respectively. The shaded gray area represents the many-body quasiparticle, where the width is proportional to the peak weight. The black thick dot represents the location of the exact Kohn-Sham eigenstates. The poles of the spin-polarized mean-field solution are shown with a dashed red line. Energy units are arbitrary.

Figure 4

The eight pieces in the $(N_{\uparrow },N_{\downarrow })$ plane that must be used to perform the constrained minimization procedure leading to the exact $Q$ functional of the double-site Anderson and Hubbard models. The thick dots indicate the positions where $G$ and $G^{\mathit{KS}}$ are evaluated.

Figure 5

(a) Three-dimensional plot of the ground-state energy as a function of $(N_{\uparrow },N_{\downarrow })$ for the symmetric case with $U=4$. (b) Ground-state energy along the paramagnetic line in the symmetric case for several $U$ values. Energies are given in units of $\left|t\right|$.

Figure 6

Quasiparticle spectrum of the symmetric double-site Anderson model as a function of the electron number $N$, computed at the thick dots shown in Fig. 4. Green's functions poles for $U=1$ [(a1) and (a2)] and $U=10$ [(b1) and (b2)]. The upper and lower panels show $G_{d,\uparrow }$ and $G_{d,\downarrow }$, respectively. The energies of the many-body quasiparticles are shown as black dots, the width of which is proportional to the quasiparticle weight. The Kohn-Sham eigenstates are displayed as red dashes. The position of the one-particle HOMO level is marked by a thicker dash. Energies are given in units of $\left|t\right|$.