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In a recent letter [Phys. Rev. Lett. 131, 216401] we presented the multichannel Dyson equation (MCDE) in which two or more many-body Green's functions are coupled. In this work we will give further details of the MCDE approach. In particular we will discuss: 1) the derivation of the MCDE and the definition of the space in which it is to be solved; 2) the rationale of the approximation to the multichannel self-energy; 3) a diagrammatic analysis of the MCDE; 4) the recasting of the MCDE on an eigenvalue problem with an effective Hamiltonian that can be solved using standard numerical techniques. This work mainly focuses on the coupling between the one-body Green's function and the three-body Green's function to describe photoemission spectra, but the MCDE method can be generalized to the coupling of other many-body Green's functions and to other spectroscopies.

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Hedin's equations provide an elegant route to compute the exact one-body Green's function (or propagator) via the self-consistent iteration of a set of non-linear equations. Its first-order approximation, known as $GW$, corresponds to a resummation of ring diagrams and has shown to be extremely successful in physics and chemistry. Systematic improvement is possible, although challenging, via the introduction of vertex corrections. Considering anomalous propagators and an external pairing potential, we derive a new self-consistent set of closed equations equivalent to the famous Hedin equations but having as a first-order approximation the particle-particle (pp) $T$-matrix approximation where one performs a resummation of the ladder diagrams. This pp version of Hedin's equations offers a way to go systematically beyond the $T$-matrix approximation by accounting for low-order pp vertex corrections.

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The Bethe–Salpeter equation (BSE) is the key equation in many-body perturbation theory based on Green's functions to access response properties. Within the GW approximation to the exchange-correlation kernel, the BSE has been successfully applied to several finite and infinite systems. However, it also shows some failures, such as underestimated triplet excitation energies, lack of double excitations, ground-state energy instabilities in the dissociation limit, etc. In this work, we study the performance of the BSE within the GW approximation as well as the T-matrix approximation for the excitation energies of the exactly solvable asymmetric Hubbard dimer. This model allows one to study various correlation regimes by varying the on-site Coulomb interaction U as well as the degree of the asymmetry of the system by varying the difference of potential Δv between the two sites. We show that, overall, the GW approximation gives more accurate excitation energies than GT over a wide range of U and Δv. However, the strongly correlated (i.e., large U) regime still remains a challenge.

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The subject of the thesis focuses on new approximations studied in a formalism based on a perturbation theory allowing to describe the electronic properties of many-body systems in an approximate way. We excite a system with a small disturbance, by sending light on it or by applying a weak electric field to it, for example and the system "responds" to the disturbance, in the framework of linear response, which means that the response of the system is proportional to the disturbance. The goal is to determine what we call the neutral excitations or bound states of the system, and more particularly the single excitations. These correspond to the transitions from the ground state to an excited state. To do this, we describe in a simplified way the interactions of the particles of a many-body system using an effective interaction that we average over the whole system. The objective of such an approach is to be able to study a system without having to use the exact formalism which consists in diagonalizing the N-body Hamiltonian, which is not possible for systems with more than two particles.

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We present the multi-channel Dyson equation that combines two or more many-body Green's functions to describe the electronic structure of materials. In this thesis, we use it to model photoemission spectra by coupling the one-body Green's function with the three-body Green's function and to model neutral excitation by coupling the two-body Green's function with the four-body Green's function . We demonstrate that, unlike methods using only the one-body Green's function, our approach puts the description of quasiparticles and satellites on an equal footing. We propose a multi-channel self-energy that is static and only contains the bare Coulomb interaction, making frequency convolutions and self-consistency unnecessary. Despite its simplicity, we demonstrate with a diagrammatic analysis that the physics it describes is extremely rich. Finally, we present a framework based on an effective Hamiltonian that can be solved for any many-body system using standard numerical tools. We illustrate our approach by applying it to the Hubbard dimer and show that it is exact both at 1/4 and 1/2 filling.

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Sujets

Atom laser Chaos Coulomb presssure Nuclear Deposition dynamics Time-dependent density-functional theory Electronic properties of metal clusters and organic molecules Oxyde de nickel Electron emission Neutron Induced Activation Au-delà du champ moyen Fission Photon interactions with free systems Activation neutronique Angle-resolved photoelectron spectroscopy Energy spectrum Neutronique Théorie de la fonctionnelle de la densité 3620Kd Damping Relaxation Optical response Agregats Electron-surface collision Landau damping Dissipation Inverse bremsstrahlung collisions Ionization mechanisms Green's function Metal cluster TDDFT Hierarchical model Electronic excitation FOS Physical sciences Instabilité Molecules 3115ee Multirefence methods Electronic properties of sodium and carbon clusters Molecular dynamics Fonction de Green Numbers 3360+q Matel clusters Nanoplasma Explosion coulombienne Méthode multiréférence Instability Correction d'auto-interaction Effets dissipatifs Electron correlation Aggregates Greens function methods Extended time-dependent Hartree-Fock Semiclassic Collision frequency Neutronic Lasers intenses MBPT Dissipative effects Electric field Corrélations dynamiques Mean-field Nucléaire Interactions de photons avec des systèmes libres Monte-Carlo Embedded metal cluster Metal clusters Ar environment Hubbard model Méthodes des fonctions de Green Modèle de Hubbard Deposition Collisional time-dependent Hartree-Fock Density Functional Theory Diffusion Hierarchical method CAO Irradiation moléculaire Matrice densité Approximation GW Photo-Electron Spectrum High intensity lasers Coulomb explosion Laser Clusters Electronic emission Nickel oxide Champ-moyen 3640Cg GW approximation Méchanismes d'ionisation Agrégats Dynamique moléculaire Environment Density-functional theory Photo-electron distributions Corrélations Molecular irradiation Dynamics Corrélation forte

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