Scrambling of Hartree-Fock levels as a universal Brownian-motion process

We study scrambling of the Hartree-Fock single-particle levels and wave functions as electrons are added to an almost-isolated diffusive or chaotic quantum dot with electron-electron interactions. We use the generic framework of the induced two-body ensembles, where the randomness of the two-body interaction matrix elements is induced by the randomness of the eigenfunctions of the chaotic or diffusive single-particle Hamiltonian. After an appropriate scaling of the number of added electrons, the scrambling behaviors of both the Hartree-Fock levels and wave functions are each described by a universal function. These functions can be derived from a parametric random-matrix process of the Brownian-motion type. An exception to this universality occurs when an empty level gets filled, in which case scrambling is delayed by one electron. An explanation of these results is given in terms of a generalized Koopmans’ approach.

Figure 1

The average HF level density $\rho \left(ϵ\right)$ for $n=10$ electrons and $u=0.1$ (histogram) is compared with an empirical fit (see text, solid line) and Wigner’s semicircle ($u=0$, dashed line).

Figure 2

Top panels: Squared wave-function overlap $o_{\alpha }^{\left(\delta n\right)}$ versus number $\delta n$ of added electrons for several HF levels $\alpha$. Bottom panels: Variance ${\sigma }^2(\alpha ,\delta n)$ of the change in the HF energy of level $\alpha$ versus $\delta n$. Steps are observed across the HF gap. The left (right) panels correspond to an interaction parameter $u=0.05$ $(u=0.1)$. In all cases, the starting number of electrons is $n=10$.

Figure 3

Same data as in Fig. 2 but as a function of $\delta n_{\mathrm{eff}}$ [see Eq. (4)]. The solid lines are predicted from a continuous Gaussian process with exponent $\eta =1$. To avoid overlap, the curves are shifted with respect to each other. The agreement is quite remarkable except when an empty level becomes filled (i.e., for $\alpha =14,18$).

Figure 4

The addition of electrons to a quantum dot is analogous to Brownian motion. Shown are three HF single-particle levels ${ϵ}_{\alpha }^{\left(n\right)}$ versus the number $n$ of electrons for $u=0.1$ (right) and $u=0.01$ (left). Notice the finer scale on the left.