Exchange-correlation potentials for multiorbital quantum dots subject to generic density-density interactions and Hund's rule coupling

By reverse engineering from exact solutions, we obtain Hartree-exchange-correlation (Hxc) potentials for a double quantum dot subject to generic density-density interactions and Hund's rule coupling. We find ubiquitous step structures of the Hxc potentials that can be understood and derived from an analysis of stability diagrams. We further show that a generic Hxc potential can be decomposed into four basic potentials which allows for a straightforward parametrization and paves the road for the construction of Hxc potentials for interacting multiorbital systems. Finally, we employ our parametrization of the Hxc potential in density functional theory calculations of multiorbital quantum dots and find excellent agreement with exact many-body calculations.

Figure 1

Panels (a)–(c) (constant interaction model, CIM): Stability diagram (a) and Hxc potentials for orbitals 1 and 2 [panels (b) and (c), respectively] of the double quantum dot for $U_1=U_2=U_{12}$. Panels (d)–(f) (regime I): Stability diagram (d) and Hxc potentials for orbitals 1 and 2 [panels (e) and (f), respectively] of the double quantum dot for $U_1=2.5U_{12}, U_2=3U_{12}$. All energies in units of the smallest interaction ($U_{12}$).

Figure 2

Panels (a)–(c) (regime II): Stability diagram (a) and Hxc potentials for orbitals 1 and 2 [panels (b) and (c), respectively] of the double quantum dot for $U_2=4U_1$ and $U_{12}=2U_1$. Panels (d)–(f) (regime III): stability diagram (d) and Hxc potentials for orbitals 1 and 2 [panels (e) and (f), respectively] of the double quantum dot for $U_2=2U_1$, and $U_{12}=2.5U_1$. All energies in units of the smallest interaction ($U_1$).

Figure 3

Panels (a), (b) stability diagram (a) and Hxc potential of orbital 2 (b) for the skew interaction ${\mathcal{V}}_{\mathrm{skew}}=\frac{U}{2}{\widehat{n}}_2(\widehat{N}-1)$. The structure of the Hxc potential for orbital 1 is the same as for orbital 2 but the step heights are half those of orbital 2 (0, $U/2, U$). Panels (c), (d) stability diagram (c) and Hxc potential of both orbitals (d) for the interorbital interaction ${\mathcal{V}}_{\mathrm{inter}}=U{\widehat{n}}_1{\widehat{n}}_2$. All energies in units of $U$ in both cases.

Figure 4

Schematic representation of the four basic Hxc potentials for building the generic potentials for all three regimes. (a) Hxc potential for CIM interaction $\frac{U}{2}\widehat{N}(\widehat{N}-1)$. (b) Hxc potential for inter-orbital interaction $U{n}_1{n}_2$. (c), (d) Hxc potential for intraorbital (i.e., SSM) interactions $U{n}_{\alpha \uparrow }{n}_{\alpha \downarrow }$. (e), (f) Hxc for the skew interaction $\frac{U}{2}{\widehat{n}}_2(\widehat{N}-1)$.

Figure 5

Effect of Hund's rule coupling on (a) stability diagram and (b) Hxc potential of orbital 1 for CIM interaction plus Hund's rule coupling, ${\mathcal{V}}_{\mathrm{int}}=\frac{U}{2}\widehat{N}(\widehat{N}-1)+{\mathcal{V}}_{\mathrm{Hund}}$ for $U=2J_H$. Here, due to symmetry, the Hxc potential for orbital 2 can simply be obtained by reflection along the $n_1=n_2$ line. All energies in units of $J_H$.

Figure 6

Comparison of parametrized and exact Hxc potentials as a function of $N=n_1+n_2$ for three basic interactions: (a) CIM interaction ($U_1=U_2=U_{12}>0$); (b) interorbital interaction ($U_{12}>0$ and $U_1=U_2=0$); (c) and (d) skew interaction ($U_2=2U_{12}>0$ and $U_1=0$); all energies in units of the smallest nonzero interaction ($U_{12}$).

Figure 7

Density $\mathit{n}=(n_1,n_2)$ as function of the gate voltage $v_g$ for different temperatures when $U_1=U_2=3U_{12}>0$ (regime I). The DFT result (solid line) is practically on top of the GCE result (dashed line) in the low temperature regime. All energies in units of the smallest interaction $U_{12}$.

Figure 8

Density $\mathit{n}=(n_1,n_2)$ as a function of the gate voltage $v_g$ for $U_1=3U_{12}, U_2=2.5U_{12}$ (regime I) for (a) low and (b) high temperatures. The inset of panel (b) shows the path in the $n_1-n_2$ plane as the gate is varied for different temperatures. The grey lines show the steps of the CIM and SSM terms that appear in the Hxc potentials. All energies in units of the smallest interaction $U_{12}$.

Figure 9

(a) Comparsion of the evolution of the density $\mathit{n}=(n_1,n_2)$ with the gate voltage $v_g$ in regime II ($U_2=2U_{12}=4U_1$) and regime III ($U_2=U_{12}=4U_1$). (b) Comparison of density evolution for two different values of the splitting $\delta v$ in regime II ($U_{12}=2U_1$). The inset shows the different paths in the density plane. $\beta =20/U_1$ everywhere. All energies in units of $U_1$.

Figure 10

Local occupations for the triple (a), (b) and quadruple (c), (d) QDs as function of the gate voltage. In the left panels, the QD levels are taken at particle hole ($v_{\alpha }={\epsilon }_{\alpha }^{*}+v_g$) and in the right the impurity level is set to zero ($v_{\alpha }=v_g$). $\beta =20/U^{\prime }$ and $U_1=5U^{\prime }, U_2=4U^{\prime }, U_3=3U^{\prime }, U_4=2U^{\prime }$. All energies in units of $U^{\prime }$.