Landau Fermi-liquid picture of spin density functional theory: Strutinsky approach to quantum dots

We analyze the ground-state energy and spin of quantum dots obtained from spin density functional theory (SDFT) calculations. First, we introduce a Strutinsky-type approximation, in which quantum interference is treated as a correction to a smooth Thomas-Fermi description. For large irregular dots, we find that the second-order Strutinsky expressions have an accuracy of about 5% of a mean level spacing compared to the full SDFT and capture all the qualitative features. Second, we perform a random matrix-theory/random-plane wave analysis of the Strutinsky SDFT expressions. The results are statistically similar to the SDFT quantum dot statistics. Finally, we note that the second-order Strutinsky approximation provides, in essence, a Landau Fermi-liquid picture of spin density functional theory. For instance, the leading term in the spin channel is simply the familiar exchange constant. A direct comparison between SDFT and the perturbation theory derived “universal Hamiltonian” is thus made possible.

Figure 1

(Color online) The functions $v_a(n,0)$ and $v_b(n,0)$, which define the functional second derivative of ${\mathcal{E}}_{\mathrm{xc}}[n^{\alpha },n^{\beta }]$ [see Eq. (34)], as a function of the parameter $r_s=1∕\sqrt{\pi a_0^{2}n}$.

Figure 2

(Color online) Particle density for the parameters [a4] for a system of $N=200$ electrons $\left(N_{\alpha }=N_{\beta }=100\right)$. From top to bottom: $n_{\mathrm{TF}}\left(\mathbf{r}\right)$, $n_{\mathrm{KS}}\left(\mathbf{r}\right)$, $\delta n\left(\mathbf{r}\right)$, and ${\tilde{n}}_{\mathrm{osc}}\left(\mathbf{r}\right)$. Note that $n_{\mathrm{TF}}$ is a smooth approximation to $n_{\mathrm{KS}}$ and that $\delta n\left(\mathbf{r}\right)$ and ${\tilde{n}}_{\mathrm{osc}}\left(\mathbf{r}\right)$ are very similar.

Figure 3

(Color online) Diagonal cut of particle densities $\delta n\left(\mathbf{r}\right)$ (solid) and ${\tilde{n}}_{\mathrm{osc}}\left(\mathbf{r}\right)$ (dashed), in units of the average Thomas-Fermi density $n_{\mathrm{TF}}$ inside the dot. Upper panel: parameters [a1] $\left(r_s\simeq 0.3\right)$. Lower panel parameters [a4] $\left(r_s\simeq 1.3\right)$. The dot contains $N=200$ electrons $\left(N_{\alpha }=N_{\beta }=100\right)$. The thin dashed lines correspond to ${\tilde{n}}_{\mathrm{osc}}\left(\mathbf{r}\right)$ from a less accurate choice of the Thomas-Fermi kinetic energy term, namely, $\eta =1.0$ instead of $0.25$.

Figure 4

(Color online) Difference between Strutinsky and Kohn-Sham ground-state energies, in units of the mean level spacing, as a function of the number of electrons, for the high density dot [a1] $\left(r_s\simeq 0.3\right)$. Upper panel: $E_{\mathrm{ST}}\left[N\right]$ obtained from Eq. (23). Lower panel: $E_{{\mathrm{ST}}^{*}}\left[N\right]$ obtained from Eq. (25). In both cases, the difference shows small fluctuations (a few percent) about a linear secular trend (dashed line is fit).

Figure 5

(Color online) Difference between Strutinsky and Kohn-Sham ground-state energies as in Fig. 4 but now with the secular trend removed. Solid: all ground states. Dots: ground states without significant spin contamination. Note the excellent agreement in the case of $E_{\mathrm{ST}}$, obtained with the intermediate result Eq. (23) (upper panel)—of order 1%—when spin contamination is not present. In the case of the “genuine” Strutinsky approximation $E_{{\mathrm{ST}}^{*}}$ [i.e., using Eq. (25)], the agreement is still very good (lower panel).

Figure 6

Spin contamination $\delta S\left[N\right]$ as a function of the number of particles. The spin contamination is infrequent at high density ([a1], upper panel), but becomes frequent and substantial for $r_s\simeq 1.3$ ([a4], lower panel).

Figure 7

(Color online) Difference between Strutinsky and Kohn-Sham ground-state energies, in units of the mean level spacing, as a function of the number of electrons, for the $r_s\simeq 1.3$ dot [a4]. Upper panel: $E_{\mathrm{ST}}\left[N\right]$; lower panel: $E_{{\mathrm{ST}}^{*}}\left[N\right]$. The linear secular trend (dashed line is a fit) is now significantly larger than for [a1] (compare with Fig. 4), but the fluctuation about this trend remains small (of order 5%).

Figure 8

(Color online) Spin and peak spacing distributions for the cases [a1] (left column) and [a4] (right column). Solid: even $N$; dashed: odd $N$. The statistics are obtained for $N=120–200$ with $(\lambda ,\gamma )=(0.53,0.2),(0.565,0.2),(0.6,0.1),(0.635,0.15)$ and $(0.67,0.1)$. From top to bottom: Kohn-Sham, intermediate [Eq. (23)], and genuine [Eq. (25)] Strutinsky approximation. Agreement between the three methods is excellent, of course, for [a1]. But even for [a4] where individual energies may be in error, the agreement of the distributions is very good.

Figure 9

(Color online) Average over the Fermi circle of the eigenvalues of the screened and bare SDFT interactions [in units of $2∕N\left(0\right)$] as a function of $r_s={\left(\pi a_0^{2}n\right)}^{-1∕2}$. Dark: charge channel. Lighter color (green online): spin channel. Dashed: bare interaction $⟨{\lambda }_{c,s}^{\text{bare}}⟩$ (with a cutoff of the momentum at $q\simeq 1∕L$ in the charge channel). Solid: screened interaction $⟨{\lambda }_{c,s}⟩$ with $\eta =0.25$. Thin dot-dashed: same but with $\eta =0$. Since the $q$ dependence of ${\lambda }_s$ is entirely due to the ${\mathcal{T}}_{\mathrm{TF}}^{\left(1\right)}$ correction to the Thomas-Fermi kinetic energy functional, it therefore disappears in this latter case. The interaction in the charge channel is, of course, dramatically decreased by screening; in contrast, screening increases the magnitude of the interaction in the spin channel.

Figure 10

(Color online) Mean value of the gain in interaction energy $-\left[\Delta E^{\left(2\right)}(S_z+1)-\Delta E^{\left(2\right)}\left(S_z\right)\right]$, as computed from Eq. (50), (thick lines) and of the mean one-particle energy cost (thin horizontal lines) associated with flipping the spin of one particle in the quantum dot. Solid: $S_z=0$; dot-dashed $S_z=1∕2$; Dashed: $S_z=1$.

Figure 11

(Color online) Comparison between the analytical RPW predictions [Eqs. (48, 51)] and numerical calculations of the mean and variance of $M_{ij}$’s. The wave functions used in the numerical calculations are eigenfunctions of the effective Thomas-Fermi potential with $N=200$ in the quartic oscillator system. Lines (points) correspond to analytical (numerical) results, with the solid (circle) for $M_{i,j}^{\sigma ,\sigma }$, short dashed (square) for $M_{i,j}^{\alpha ,\beta }$, long dashed (uptriangle) for $M_{i,i}^{\sigma ,\sigma }$, and dot dashed (down triangle) for $M_{i,i}^{\alpha ,\beta }$.

Figure 12

(Color online) Peak spacing distributions for the RMT/RPW model corresponding to $r_s=0.3$ (left) and $1.3$ (right). Solid: even $N$, dashed: odd $N$. Inset: corresponding spin distribution.

Figure 13

(Color online) Same as Fig. 12b, but with a residual interaction modeled by a perturbative calculation in the interaction potential $V_{\mathrm{RPA}}(\mathbf{r}-{\mathbf{r}}^{\prime })$ whose parameters are set by Eqs. (64, 65).