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, such as ground-state energies, one particle excitation energies, densities and other measurable quantities. Therefore the development of approximate methods to calculate the one-body GF has been an active research topic in many-body physics since the 60's, and many routes have been explored in order to find increasingly accurate GFs.
A very popular class of methods is based on the solution of an integral equation for the GF containing an effective potential, the so-called self-energy, which needs to be approximated. The well-known GW approximation belongs to this class; this approximation is the method of choice for calculating band structures, but it also shows several shortcomings, such as the wrong description of satellites in photoemission spectra, in particular in so-called strongly-correlated materials. Therefore more refined levels of approximations are needed to keep the pace with the advances made in experiment. Recently much progress has been made in this direction both by going beyond standard methods and also exploring completely novel routes to calculate GFs. A new wave of original ideas, understanding, and solutions, has pervaded the field. It is therefore timely to gather these new concepts in a workshop for the first time.
Self-energy based methods are standard tools in Many-Body Perturbation Theory for the calculations of Green’s functions. A good starting point to make approximations to the self-energy is to use Hedin’s equations, which are often solved within the so-called GW approximation (GWA) to the self-energy , where G is the one-particle Green’s function and W the dynamically screened Coulomb potential. Within this approximation one neglects so-called vertex corrections, which take into account the fermionic nature of the system, and one treats the system and its response classically. This approximation works well for systems where the screening is important; instead in systems which show atomic-like physics, such as transition-metal oxides, the GWA shows difficulties or even failures [2-7] More accurate approximations are then needed, in order to describe the wealth of emergent new physics.
One way to go beyond the standard methods is to try to correct some basic shortcomings that plague them. As a first step this requires a clear understanding of the failures of standard approaches. Tremendous progress has been made in recent years to understand why current approximations fail in certain situations. These failures are related to self-screening (the unphysical screening of an electron by itself) [2-4], lack of full self-consistency [8-10], inconsistent treatment of dynamical effects [11-14], inability to capture strong correlation at (near-)degeneracy [3,15-17], to name a few. The solution of these problems lies in the inclusion of vertex corrections beyond the GWA in the self-energy [3,4,18].
Besides a constructive inclusion of vertex corrections thanks to a better physical understanding of the fundamental equations of Many-Body Perturbation Theory, more attention has also been paid to their mathematical structure. This has been done, for example, by studying these equations in models that are exactly solvable, such as the 1-point model [19-23] or the Hubbard model [3,5,6,18]. This understanding helps greatly to make physically motivated approximations with which to calculate real systems, but also elucidates the origin of possibly unphysical results that one can obtain in calculations. For example, the existence of multiple solutions to Hedin’s equations is a hot topic and only very recently raised the interest that it deserves. The question of how many well-behaved solutions exist, how to pick the physical one, and if a self-consistent scheme converges to it [22-24] are still open issues for the MBPT community and now these questions start to be answered.
Finally, besides the efforts devoted to finding better approximations to the self-energy, novel routes are explored in order to directly approximate the Green’s function itself [22,23]. This strategy has, for example, allowed a consistent derivation  of the so-called cumulant expansion [7,25,26], originally obtained as an exact solution to a simplified model to describe core spectroscopy , in an ab initio and valence framework.This allowed for the correct description of multiple satellites in photoemission spectra , something that was not possible with standard methods. Moreover, this new derivation of the cumulant expansion is general and can be used to go beyond this approximation. Focus is also put on the direct calculation of the GF as solution of a functional differential equation. This has been solved for a simplified model which gave many important insights for the solution of the full functional problem [22,23].