Laboratoire de Physique Théorique
Toulouse - UMR 5152

The many-body problem, i.e. the solution of the Schroedinger equation in interacting many-body systems, is a formidable problem in condensed matter and nuclear physics, like in quantum chemistry, so far unsolved despite considerable efforts. Since the Hartree-Fock method, different theories have been developed : approaches relying on the wave function, like quantum Monte Carlo or quantum chemistry methods, or on the electronic density, like density-functional theory (DFT) in its static or time-dependent (TDDFT) form or, finally, quantum field theoretical, Green function based methods. While exact in principle, in practice all of them rely on approximations and recipes whose validity are difficult to judge. Validation of them against exact solutions in realistic models or simple real systems, is an unavoidable step for further improvement.

Helium atom is the simplest many-electron (just only two electrons) system in nature, the only for which we know the really exact (to any degree of accuracy) 1929 Hylleraas solution. We will compare the exact excitations spectrum of helium to the approximated solution of many of the approaches listed above, though with a focus in particular to Green function methods in condensed matter (e.g. the Bethe-Salpeter equation and the GW approximation on the self-energy) and in nuclear physics (standard and self-consistent RPA). This will allow to evaluate their validity, pros and cons, and establish absolute errors.