The Luttinger-Ward method applied to the 2D Coulomb gas

We review the Luttinger-Ward method and show how one can combine it with the thory of Baym and Kadanoff in order to get conserving approximations in many-body perturbation theory. We solve the variation procedure numerically using an ansatz for the selfenergy in the case of the two-dimensional Coulomb problem. Comparison with results of the traditional iteration of the selfconsistency equation yields satisfactory agreement for parameters where the iteration converges. The extremalization of the Luttinger-Ward functional works beyond that point.