University of Virginia

The SCF method gives an approximate treatment of electron-electron interactions in terms of charging capacitors. The interaction is, however, quite complex since the electrons do not act independently but often correlate their configurations to reduce the Coulomb cost of charging. Such interactions require keeping track of all many-electron configurations, and not just focusing on a single conducting electron moving in a self-consistent background field. As long as the broadening g is much larger than the charging energy U₀, as in large objects, such configurations average out. For quantum dots with weak coupling to contacts, however, many-body effects dominate.
In the many-body approach, `levels' are interpreted as differences between total energies of the neutral (N-electron) and cationic (N-1) or anionic (N+1) electron systems. Furthermore, the electrons do not need to be in the equilibrium ground state of a given N-electron many-body configuration, but can be thermally promoted to excited states. Transport is driven by the evolving probabilities of the many- body states -- an incoming electron takes the whole system from an N to an N+1 electron state, while a removed electron takes it from N to N-1. By setting up a rate equation in this many-body (Fock) space, we can calculate the steady-state probabilities of each many-electron configuration. This allows us to compute the expected electron number and the terminal currents (keeping track of addition minus removal processes at that terminal).
The many-body current-voltage characteristic differs from the SCF result even for simple systems, as long as U₀ > g. For instance, the current carried by the first incoming spin could be different from the second spin, as some of the second spin configurations (parallel to the first one) are blocked by Pauli exclusion. The combination of Coulomb repulsion and Pauli exclusion thus generates novel states that cannot be described as independent products of one-electron states. A well-known example of such a correlated state is a ferromagnet, where the spins all align and are thus not independent. Other examples include antiferromagnets, superconductors, fractional quantum Hall states, excitons and Kondo singlets.