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 of both the HF levels and wave functions is described by a universal function each. 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 just 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.