Solution of the many-body problem: a step back to progress – PROGRESS

Electronic correlation is the source of a wide range of interesting phenomena, including superconductivity, the fractional quantum hall effect, or van der Waals dispersion forces. Although the underlying interaction (the Coulomb interaction) is « simple » and well understood, a unifying framework is still missing that would allow us to describe, analyze, understand and predict all those phenomena on the same footing. In this project we propose to explore a completely new path to the solution of the many-body problem. The idea is to go a step back in the derivation of the Kadanoff Baym equations that determine the one-and two-particle Green's functions, on the level of the equation of motion of the one-particle Green's function. This equation can be transformed into a set of coupled multi-dimensional non-linear first-order differential equations, which we suggest to solve approximatively in at least two ways: numerically, by discretization of the problem, and semi- analytically, with a suitable ansatz. This work will include the solution of Hubbard clusters performed with the proposed approach, the design of an optimized analytical ansatz for the solution, containing several parameters, a derivation of an approximate vertex correction from the above results, and an ab initio implementation of the method. To the best of our knowledge, such an approach has never been tried, at least during the last 20 years where computer resources have become strong enough to make an attempt at least in principle feasible.