  The possibility of finding multistability in the density and current of an
interacting nanoscale junction coupled to semi-infinite leads is studied at
various levels of approximation. The system is driven out of equilibrium by an
external bias and the non-equilibrium properties are determined by real-time
propagation using both time-dependent density functional theory (TDDFT) and
many-body perturbation theory (MBPT). In TDDFT the exchange-correlation effects
are described within a recently proposed adiabatic local density approximation
(ALDA). In MBPT the electron-electron interaction is incorporated in a
many-body self-energy which is then approximated at the Hartree-Fock (HF),
second-Born (2B) and GW level. Assuming the existence of a steady-state and
solving directly the steady-state equations we find multiple solutions in the
HF approximation and within the ALDA. In these cases we investigate if and how
these solutions can be reached through time evolution and how to reversibly
switch between them. We further show that for the same cases the inclusion of
dynamical correlation effects suppresses bistability.
