# Scientific goals and main achievements

### Fundamental understanding

Many-body effects influence electronic spectra in quantitative and often even qualitative ways.
Since our understanding is mainly based on an independent-electron picture, it is a challenge to truly understand, and be able to explain these effects, beyond the raw data produced by computer codes.

• The aim of the group is to develop methods that help understanding and facilitate analysis. The group has pointed out, and used, various connections between apparently distinct approaches, such contributing to their understanding.
In particular, it has designed a way to derive the exchange-correlation kernel of TDDFT from the Bethe-Salpeter equation of MBPT, and proposed further approximations to the result (Sottile et al. PRL 91, 056402 (2003)).
• A crucial goal is also to elucidate the fundamental mechanism in the theories that are used, point out their limitations and ways to overcome them. The functional methods that are developed are by definition very compact, since they rely on the fact that many degrees of freedom are integrated, with respect to the full many-body wavefunction. This helps computation, but involves the use of effective potentials or interactions that may display the pertinent physics in a quite indirect way. The group has explained how non-locality in the Green’s function approaches transposes into frequency-dependence in approaches based on local potentials or kernels, such as TDDFT (Gatti et al. PRL 99, 057401 (2007)). It has also explained in which way the widely used Green’s function functionals, in particular the functional of Luttinger and Ward, which is considered to be exact, may give rise to a wrong self-energy at large interaction strength, owing to the absence of a proper definition of the domain of the functionals that are involved (Tarantino et al., PRB 96, 045124 (2017)).

### More physics

Since many-body effects are complicated to understand and cumbersome to compute, many approximations have to be made. In some cases, this completely suppresses interesting physical effects. The main objectives are:

• to develop methods that help understanding and predicting pieces of physics that were previously neglected or out of reach. The group is one of the pioneers for the ab initio description of excitonic effects in semiconductors and insulators. The first ab initio solution of the Bethe-Salpeter equation (BSE) for a small cluster dates back to 1995 and was born in Palaiseau, followed by the first ab initio calculation of excitonic binding energies from the BSE for an insulator in 1997 in Palaiseau by Albrecht et al.„ and by the pioneering calculations of spectra for solids, in parallel with two different groups, in 1998 (see references in (Onida et al. RMP 74, 601 (2002)). This forefront expertise has been maintained and is further developed until today, from the description of excitonic effects on plasmons in 2001, to the extension of the BSE to describe exciton dispersion in 2013 (Gatti and Sottile, PRB 88, 155113 (2013)), and today’s ongoing works on excitonic effects in coherent inelastic x-ray scattering (PhD thesis Igor Reshetnyak) or dynamical effects in charge excitation spectra, beyond the standard BSE.
• to put also already existing approaches on new solid grounds, situate them using today’s language, and make them accessible. Many-body theory has a long history, and the group is aware of the fact that it stands on the shoulders of giants. Many interesting ideas were developed in the past and often applied to model systems. However, it is sometimes not clear how these approaches fit into the currently used frameworks, and to which extent model ideas and results can be transposed to the ab initio framework. The group has re-derived the cumulant form of the one-body Green's function from the functional differential form of the Dyson-Schwinger equation, such elucidating the underlying approximations, the link to the commonly used GW approximation, and the reasons for which the cumulant approximation is superiour to GW. Ab initio cumulant calculations had already been performed earlier by Aryasetiawan et al. (Phys. Rev. Lett. 77, 2268 (1996)), but the group’s explanations and prototype application to bulk silicon (Guzzo et al., PRL 107, 166401 (2011)) have attracted a rapidly growing community and launched a new research effort concerning plasmon satellites in the spectra of metals and insulators, pursued also in the group (Zhou et al, JCP 143, 184109 (2015)).
• to make new kinds of spectroscopy accessible to ab initio calculations. One important class of spectroscopies relies on non-linear optics. It constitutes a way to explore materials in a non-destructive way, but it is difficult to interpret, and theory is very much needed. TDDFT is an in principle very efficient framework for optics, also thanks to developments of the group [Luppi2010], but it has been used very little to explore the non-linear response of solids. The group has developed theory and a computer code to calculate second harmonic generation (SHG) in periodic systems. These developments give access to problems such as Electric Field Induced SHG, non-linear optics of surfaces and interfaces (Tancogne-Dejean et al. PRB 94, 125301 (2016)), and the search for new materials for non-linear optics (Cazzanelli et al. Nat. Mat. 11, 148 (2012)).

### Explorative

Pushing traditional methods further and further may not always be a winning strategy. For this reason, the group explores completely new ways to deal with certain questions; these new ways may, or may not, lead to breakthrough in practice, but they always lead to new insight.

• In general, perturbation theory is useful when the lowest orders are sufficient to obtain good results. When this is not true, alternatives must be searched for. This also holds for many-body perturbation theory, where, for example, the
description of satellite structures in spectra poses problems to perturbative approaches (Zhou et al, JCP 143, 184109 (2015)). The idea is to solve the functional differential Schwinger-Dyson equation for the one-body Green’s function approximately, instead of expanding it into diagrams. The group has worked on the fundaments of this idea, shining new light on the problem of multiple solutions of the many-body equations (Stan et al. NJP 17, 093045 (2015)). It has derived a systematic way to solve the equations iteratively, the above mentioned cumulant Green’s function being the solution in the simplest approximation (Marilena Tzavala, PhD thesis).
• Density Functional Theory has a huge impact in practice not least because of the introduction of the Kohn-Sham auxiliary system. However, most observables, besides the density, cannot be obtained exactly from this auxiliary system, even in principle. The idea is to generalize the concept of auxiliary systems to arbitrary observables, such maximizing the efficiency of the calculations (Gatti et al. PRL 99, 057401 (2007)). A new auxiliary system for the description of photoemission has been proposed, and the group has recently shown that this is indeed a promising approach (Vanzini et al., EPJB 91, 192 (2018)).
• Another reason for the success of DFT lies in approximations such as the Local Density Approximation (LDA): high-level Quantum Monte Carlo calculations of correlation energy densities have been performed once and forever in the homogeneous electron gas by Ceperley and Alder, and these results are “recycled” over and over in calculations for real materials, by connecting the density of the gas to the local density of the material. The idea is to generalize this “connector” approach to other observables, and to make it in principle exact. Indeed, the LDA is a rough approximation, but the group has proposed a scheme where the connection between the gas (or any other model system) and the real material appears as an in principle exact concept. This allows for the design of systematic approximations to
this connector. First results are very promising (Panholzer et al. PRL 120, 166402 (2018)). The approach should lead to a general ecologic approach, where high-level calculations are performed once and forever in a model system, stored and made freely available, and where connectors of increasing quality will allow to obtain results for real materials with a minimum of resources.

### Efficient approximations

Real materials are complex, and many-body effects are cumbersome to compute. The group continuously searches for new efficient and reliable approximations.

• The sum over a huge number of empty states is one of the bottlenecks in self-energy calculations. The group has proposed an Efficient Energy Technique (EET), where a range of transition energies is replaced by one suitably chosen effective energy, which allows one to use the closure relation instead of summing over empty states (Berger et al. PRB 82, 041103(R) (2010)).
• Excitonic effects are difficult to capture in TDDFT, if one does not allow for extremely complex and difficult to compute exchange-correlation kernels. The group has proposed a series of approximations for the kernel that capture important qualitative, and sometimes quantitative, features (Onida et al. RMP 74, 601 (2002), Rigamonti et al. PRL 114, 146402, (2015)).
• Assemblies of nano-objects with weak interaction in the ground state may interact more strongly in excited states, where monopoles or dipoles are created. This would require the simultaneous calculation of a huge number of atoms.
The group has developed a building block approach'' to evaluate the polarizability of complex objects starting from the polarizability of simpler blocks. This applies to combinations of single objects, and to the deformation of objects, such as the construction of spectra of nanotubes starting from graphene (Kramberger et al. PRL 100, 196803 (2008)) and (Mohn et al. PRB 97, 235410 (2018)) (see also Ralf Hambach’s thesis).
• The calculation of the spectra of isolated objects using periodic boundary conditions poses some subtle problems. The group has proposed ways to overcome these problems, also in the presence of the effects of interactions, such as local field effects (Tancogne-Dejean et al. PRB 92, 245308 (2015)).

### Materials

The group is not specialized in one class of materials, but takes pleasure to investigate and collaborate on changing, challenging problems of real materials. Past or ongoing collaborations therefore span a wide range. This includes companies searching for materials for photovoltaic applications, electric transport, or transparent conducting materials, synchrotrons, or the museum for natural history in Paris. Recent examples for materials-specific results are:

• Strong correlation has become a buzzword to label many different problems of physics. The aim of the group is to investigate materials that are named as such, and to elucidate the pertinent physics. For example, the group has shown that photoemission and/or optical absorption spectra of SrVO3 and VO2 can be understood when exchange and screening are correctly taken into account, without the need for non-perturbative approaches (Gatti et al. PRL 99, 057401 (2007), Gatti and Sottile, PRB 88, 155113 (2013), Gatti et al., PRL 114, 116402 (2015)).
• While spectra in linear optics can often be understood in terms of sums of transitions, non-linear spectra are much more complicated to analyse and to predict. The aim of the group is to use the in-house developments to investigate interesting applications. Results have been obtained for surfaces, interfaces, clusters, nanotubes, and magnetic materials. In particular, it has been shown how strain and stress influence the SHG in silicon (Cazzanelli et al. Nat. Mat. 11, 148 (2012)), guiding experimentalists in the optimization of SHG efficiency.
• Excitonic effects are strongly enhanced in low dimensions, for example in molecules. It is then interesting to study molecular solids, which present both 1D and 3D characteristics. The aim of the group is to calculate optical spectra and exciton dispersion, and to understand the results using simple models. In particular the excitons in molecular solids have been predicted and analysed, in collaboration with experimentalists (Cudazzo et al. PRB 88, 195152 (2013)).
• The 2D character of layered materials and the interaction between layers are an ideal playground for research on electronic excitations. The aim of the group is to help experimentalists disentangle inter-and intra-layer effects. A recent result concerns excitonic spectra of layered, including exciton dispersion and the influence of stacking (Bourrellier et al., ACS Photonics 1, 857(2014); Koskelo et al, PRB 95, 035125 (2017); Cudazzo et al., NJP 18, 103050 (2016)).