Report on scipost_202110_00015v2
Gabriele Riva, Timothée Audinet, Matthieu Vladaj, Pina Romaniello, J. A. Berger · 2021
We present an original approach for the calculation of direct and inverse photoemission spectra from first principles.The main goal is to go beyond the standard Green's function approaches, such as the GW method, in order to find a good description not only of the quasiparticles but also of the satellite structures, which are of particular importance in strongly correlated materials.Our method uses as a key quantity the three-body Green's function, or, more precisely, its hole-hole-electron and electron-electron-hole parts.We show that, contrary to the one-body Green's function, satellites are already present in the corresponding non-interacting Green's function.Therefore, simple approximations to the three-body self-energy, which is defined by the Dyson equation for the three-body Green's function and which contains many-body effects, can still yield accurate spectral functions.In particular, the self-energy can be chosen to be static which could simplify a self-consistent solution of the Dyson equation.We also show how the one-body Green's function can be retrieved from the three-body Green's function.We illustrate our approach by applying it to the symmetric Hubbard dimer.3.3 1/2 filling 14 4 Conclusions and Outlook 17 A Derivation of the spectral representation of G 3 18 B Recovering G 1 from G 3 20 C Inversion of the Dyson equation 21 D Eigenvalues and eigenvectors of the symmetric Hubbard dimer 21 E Diagonal G e+h 3 for Hubbard dimer at 1/4 filling 23 F Diagonal G e+h 3 for Hubbard dimer at 1/2 filling 24 References 27SciPost Physics Submission self-energy is inconvenient from a practical point of view because it makes self-consistent calculations very cumbersome.Although fully self-consistent GW calculations have been performed on small atoms and molecules [3-10], there are, to the best of our knowledge, no such calculations for solids.Therefore, whenever self-consistency is important, one usually employs partial self-consistent GW methods, e.g.quasi-particle self-consistent GW , that use a static approximation to the GW self-energy [11][12][13][14][15].As a consequence, there is no self-consistent GW approach that can treat both quasiparticles and satellites in solids.We note that an alternative to solving the Dyson equation is to make an ansatz for the 1-GF, which is the strategy of the cumulant approach [16,17].When combined with GW this method has been shown to yield accurate quasiparticle energies as well as plasmon satellites [18][19][20][21][22][23][24][25][26].However, the precision of the GW plus cumulant approach for other types of satellites has still to be investigated.In this work we adopt a completely different strategy to capture the physics of both quasi-particles and satellites.In an (inverse) photoemission process a hole (electron) is created and the system will react to this extra particle, by creating electron-hole pairs.Photoemission spectroscopy could therefore be seen as a three-particle process, the electron or hole that is added plus an electron-hole pair.Therefore, we will study here the three-body Green's function (3-GF) as the fundamental quantity from which to calculate photoemission spectra.In particular, we will study the electron-hole-hole 3-GF (G ehh 3 ) and the electron-electron-hole 3-GF (G eeh 3 ) which contain all the required information about photoemission and inverse photoemission spectra, respectively.We will show that already at the level of the non-interacting 3-GF there is information about the satellites.Therefore, a static self-energy (3-SE) is sufficient to obtain both quasiparticles and satellites in the photoemission spectra.We will also demonstrate how one can retrieve the 1-GF and, therefore, the spectral function, from G ehh 3 and G eeh 3 .We illustrate these principles by studying the symmetric Hubbard dimer at 1/4 and 1/2 filling.In particular, we will show that a static approximation to the 3-SE yields excellent results for quasi-particles and satellites at weak correlation and that the results at strong correlation are still very good.Finally, we note that the three-body Green's function has been employed to describe Auger spectra [27] and to study satellite structures and the occurrence of the metal-insulator transition [28].This paper is organized as follows.In section 2 we discuss the theoretical details of the 3-GF and its link to photoemission spectra.We introduce the symmetric Hubbard dimer in section 3 and we show the results we obtained for the spectral functions.Finally, in section 4 we draw our conclusions and we discuss future perspectives.