Green's functions have always played a prominent role in many-body physics. In particular the one-body Green's function (GF) delivers a wealth of information about a physical system and many routes have been explored in order to find increasingly accurate GFs.

A popular class of methods is based on the solution of an integral equation for the GF containing an effective potential, the self-energy, which needs to be approximated. The well-known GW approximation belongs to this class [1]. Despite its success, this approximation also has several shortcomings, in particular in strongly correlated materials. Therefore more refined levels of approximations are needed. Recently much progress has been made in this direction both by going beyond standard methods and by exploring completely novel routes to calculate GFs. Some important developments are:

- One can look at smaller building blocks of G that are, in principle, easier to approximate [2] or to work with more complex resummed quantities such as in DMFT or using the 4-point vertex, from which approximations to the self-energy can be obtained [3].

- The study of the mathematical structure of the fundamental equations of Many-Body Perturbation Theory. The question of how many well-behaved solutions exist, how to pick the physical one, and if a self-consistent scheme converges to it [4,5] are still open issues for the MBPT community.

- The study of problems of perturbation expansions due to divergencies [6] and misleading convergence [7]. In these cases it is important to identify critical quantites which indicate a breakdown of perturbation theory, such as the two-particle correlation function [5,6] and to propose methods to deal with divergent series [8].

- Novel routes are explored to directly approximate the Green’s function itself using, for example, a cumulant expansion [9]. Alternatively, one can express the one-body Green's function as a continued fraction, and find a proper terminating function to include higher-order terms [10]. There are also recent efforts to combine MBPT with quantum-chemistry methods, such as coupled cluster, in order to include correlation in the one-body Green’s function [11].

In summary, a wave of original ideas, understanding, and solutions, has pervaded the field in these last years. It is therefore timely to gather them in a workshop. This has been done in 2013 and 2015, with the first two editions of this workshop. Because of the success of these workshops as well as the continuing progress in many-body Green’s function methods achieved in the last couple of years, we would like to organize the third edition in 2017 in Toulouse.

The aim of this workshop is to bring together experts in Green’s function many-body theory from all over the world who are involved in the challenging new developments in the field of Green’s functions.

One of the main goals of the workshop will be to, through a collective effort, try to find answers to the following pertinent questions:

⁃What are the effects of self-consistency and how does the final solution depend on the initial guess and on the chosen self-consistent scheme?

Besides aspects related to the implementation of a computationally feasible self-consistent scheme and the effects of self-consistency on spectral properties, there is also the fundamental question of whether a chosen self-consistent scheme will convergence to a physical solution. This is not obvious when working with complex non-linear equations and is important for developments beyond GW.

-What is the best strategy to find efficient vertex corrections?
The GWA, with the self-energy being of first order in W, is not expected to describe strong correlation. Higher orders in W could be added by iterating the equations, but this is technically difficult and is computationally expensive. Moreover, there is no guarantee that results will quickly improve or that the perturbative expansion will converge. During the workshop strategies to overcome these difficulties will be discussed.

-What is the best way to deal with the strong coupling limit?
The atomic limit is a true limitation for GW, and it is very difficult to formulate corrections that are able to capture this physics. During the workshop we will discuss the limits of perturbative self-energies in the strong correlation regime and how to deal with divergent series.

-Are there limits of thinking in terms of the self-energy?
Much effort is put in finding vertex corrections, but up to now there does not exist a well-established strategy to obtain corrections that systematically improve on GW. Therefore, formulations that directly approximate the GF
are an interesting alternative. Problems related to self-consistency, the inclusion of dynamical effects, the treatment of strong correlation, could thus be overcome. However, other problems may appear. Clarifying what are the pros and cons of this new strategy will be one of the central themes of the workshop.

⁃Which knowledge obtained from the one-body GF can be exported to the calculation
of higher-order GFs calculations?

One can recognize a common structure of the equations for the various GFs (one-body, two-body, etc.). For example, the two-body GF fulfills the so-called Bethe-Salpeter equation, which has the same structure as the Dyson equation for the one-body GF. The identification of common patterns is very useful in order to apply the same strategy to similar problems. During the workshop we will address the opportunities to transfer the lessons learned for the one-body GF to higher-order quantities.

As during previous editions of this workshop plenty of time will be allocated for discussions and exchange of ideas. We envisage to have a good mix of world experts and young promising scientists to present their recent progress in Green’s function theory.