Disordered mesoscopic systems with interactions: Induced two-body ensembles and the Hartree-Fock approach

We introduce a generic approach to study interaction effects in diffusive or chaotic quantum dots in the Coulomb blockade regime. The randomness of the single-particle wave functions induces randomness in the two-body interaction matrix elements. We classify the possible induced two-body ensembles, both in the presence and absence of spin degrees of freedom. The ensembles depend on the underlying space-time symmetries as well as on features of the two-body interaction. Confining ourselves to spinless electrons, we then use the Hartree-Fock (HF) approximation to calculate HF single-particle energies and HF wave functions for many realizations of the ensemble. We study the statistical properties of the resulting one-body HF ensemble for a fixed number of electrons. In particular, we determine the statistics of the interaction matrix elements in the HF basis, of the HF single-particle energies (including the HF gap between the last occupied and the first empty HF level), and of the HF single-particle wave functions. We also study the addition of electrons, and in particular the distribution of the distance between successive conductance peaks and of the conductance peak heights.

Figure 1

Average matrix elements ${\overline{v}}_{\alpha \beta ;\alpha \beta }$ in the HF basis vs $\alpha$ for fixed $\beta$. We show results for three values of $\beta$ $(\beta =6,12,25)$. We use the ensemble (17) with $m=30$ and for $n=10$ electrons. (a) The data are generated for $u=0.1$ and for a fixed picket-fence single-particle spectrum (equal spacings). The symbols give the numerical simulations, and the lines give the perturbative result (39). (b) As in panel (a) but for $u=0.05$ and for a fixed single-particle GOE spectrum. (c) The results are for the same ensemble as in (a), but the average is taken over the full ensemble (17), i.e., over both the single-particle GOE spectrum and the two-body interaction matrix elements of Eq. (2). The symbols represent the numerical simulations and the lines the perturbative expression (39), averaged over the single-particle GOE spectrum.

Figure 2

Top panel: Three-dimensional plot of $C_{\alpha \beta }$ [see Eq. (45)] vs $\alpha$ and $\beta$. The results shown are for $n=10$ electrons occupying $m=30$ orbitals. Bottom panel: Three-dimensional plot of the full ensemble average of $v_{\alpha \beta ;\alpha \beta }$ vs $\alpha$ and $\beta$. We use the “local” interaction ensemble with $u=0.1$ for $n=10$ electrons occupying $m=30$ orbitals. The results agree well with the perturbative expression $6u^2{C}_{\alpha \beta }$ where $C_{\alpha \beta }$ is shown in the top panel.

Figure 3

Top panel: The standard deviation of a HF matrix element $\sigma \left(v_{\alpha \beta ;\alpha \beta }\right)$ vs $u$ for the “local” interaction ensemble (14). We show results for three different types of diagonal matrix elements, corresponding to filled-filled, filled-empty, and empty-empty single-particle levels. The results shown for each type are averaged over all matrix elements of that type. Bottom panel: The fractional deviation $1-{\sigma }^2∕6u^2$ of the variance ${\sigma }^2$ of the matrix elements in the HF basis from their variance $6u^2$ in the noninteracting basis vs $u$. Same conventions as in the top panel.

Figure 4

The mean level spacing ${\Delta }_{\mathrm{HF}}$ of the single-particle HF levels vs level index $\alpha$ for $u=0.05$ (triangles), $u=0.1$ (squares), and $u=0.15$ (circles). The results shown are for $n=10$ electrons and $m=30$ single-particle states. The solid line describes the mean level spacing $\Delta$ for a GOE of dimension $m=30$. Inset: the mean level spacing in the center of the spectrum vs $u$.

Figure 5

Nearest-neighbor spacing distribution $P\left(s\right)$ vs spacing $s$ where $s$ is measured in units of the local mean level spacing of the corresponding HF levels. Top panel: The histograms show $P\left(s\right)$ for the highest two filled single-particle HF levels ${ϵ}_{n-1}^{\left(n\right)}$ and ${ϵ}_n^{\left(n\right)}$ [with $s=({ϵ}_n^{\left(n\right)}-{ϵ}_{n-1}^{\left(n\right)})∕⟨{ϵ}_n^{\left(n\right)}-{ϵ}_{n-1}^{\left(n\right)}⟩$], and for three values of $u$: (a) $u=0.05$; (b) $u=0.1$; and (c) $u=0.15$. The dashed lines give the GOE Wigner distribution. Bottom panels: same as in the top panels but for the two lowest empty single-particle HF levels ${ϵ}_{n+1}^{\left(n\right)}$ and ${ϵ}_{n+2}^{\left(n\right)}$. The values of $u$ are (d) $u=0.05$; (e) $u=0.1$; and (f) $u=0.15$.

Figure 6

Top panel: average HF gap vs $u$. The squares are the results of simulations of the full ensemble (2). The dotted-dashed line is obtained by using Eq. (46) and a spectral GOE average of the perturbative expression (39). Bottom panel: the squares depict the width $\sigma$ of a Gaussian, the convolution of which with a Wigner distribution is fitted to the gap distributions in Fig. 7. The dotted-dashed line is $\sqrt{6}u$, and the triangles (connected by the dashed line) are the standard deviations $\sigma \left(v_{n+1,n}\right)$ of the corresponding HF interaction matrix element (see text).

Figure 7

The calculated distribution of the shifted HF gap (histograms) for four values of $u$: (a) $u=0.05$; (b) $u=0.1$; (c) $u=0.15$; and (d) $u=0.25$. The gap is shifted to have the average value zero. The solid lines describe convolutions of a shifted Wigner distribution (with an appropriate $\Delta$) with a Gaussian. The width of the Gaussian is fitted. The dashed lines are the shifted Wigner distributions.

Figure 8

The spacing correlator $c_{\alpha }$ [see Eq. (47)] vs $\alpha$ for the HF spectrum of the “local” interaction ensemble with $u=0.25$ and $m=30$ (values of $\alpha$ near the edges of the spectrum are omitted). The results shown are calculated for 50 ,000 realizations of the ensemble and include statistical errors. The dashed line is the GOE value of the correlator $-0.0863\pm 0.0008$ calculated from 500, 000 realizations and averaged over $\alpha$. Inset: the correlator $c_n$ across the gap vs $u$.

Figure 9

The distributions $P\left(\ln y\right)$ vs $\ln y$, where $y$ is the square of an HF wave function component (scaled to give $\overline{y}=1$) for $u=0.05$ (triangles), $u=0.1$ (squares), $u=0.15$ (circles), and $u=0.25$ (diamonds). We use the “local” interaction orthogonal ensemble (14) with $m=30$ and $n=10$. The solid line is the finite-$m$ RMT distribution (54) and the dashed line is the Porter-Thomas distribution. A section of the graph is magnified to render more details.

Figure 10

Peak spacing distribution $P({\Delta }_2-{\overline{\Delta }}_2)$ for the same set of $u$-values as in Fig. 7. The dashed lines are (shifted) Wigner distributions.

Figure 11

The squares depict the width $\sigma$ of a Gaussian, the convolution of which with a Wigner distribution is fitted to the peak spacing distribution $P({\Delta }_2-{\overline{\Delta }}_2)$ in Fig. 10. Dotted-dashed line, triangles, and dashed line are as in Fig. 6.

Figure 12

The peak spacing distribution $P\left({\Delta }_2\right)$ (left histogram) is compared with $P({ϵ}_{n+1}^{(n+1)}-{ϵ}_n^{\left(n\right)})$ (dashed line), $P({ϵ}_{n+1}^{\left(n\right)}-{ϵ}_n^{(n-1)})$ (dotted-dashed line), and the gap distribution (right histogram). The ensemble has $u=0.1$.

Figure 13

The peak spacing distribution $P\left({\Delta }_2\right)$ (histogram) for the “local” interaction ensemble with $u=0.05$ is compared with a convolution of a Wigner distribution with a Gaussian whose width is $\sigma \left(v_{n+1,n}^{\left(0\right)}\right)$ (solid line). The dashed line describes a convolution of a Wigner distribution and a Gaussian with a width of $\sqrt{2n-1}\sigma \left(v_{n+1,n}^{\left(0\right)}\right)$. Inset: the distribution of $h_{n+1,n+1}^{\left(0\right)}-h_{n,n}^{\left(0\right)}$ (histogram) is approximately fitted by a Gaussian (solid line). The dashed line is the Wigner distribution of ${ϵ}_{n+1}-{ϵ}_n$. The number of electrons is $n=10$.

Figure 14

The distribution $P\left(\ln \widehat{\Gamma }\right)$ vs $\ln \widehat{\Gamma }$ where $\widehat{\Gamma }$ is the renormalized partial width in the HF approximation. Shown are results for $u=0.05$ (triangles), 0.1 (squares), 0.15 (circles), and 0.25 (diamonds). We use the ensemble (14) with $m=30$ and $n=10$. The solid line is the distribution (54) and the dashed line is the Porter-Thomas distribution. We have magnified a section of the graph to show more details. Inset: $\overline{\Gamma }∕\overline{\Gamma }(u=0)$ vs $u$.