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arXiv:2507.12437 · The Journal of Chemical Physics

Ground and excited-state properties of the extended Hubbard dimer from the multichannel Dyson equation

Stefano Paggi, J. A. Berger, Pina Romaniello

Published Oct 17, 2025• 2 citations
arXivHTMLPDF

Abstract

We have recently presented the multichannel Dyson equation as an alternative to the standard single-channel Dyson equation. While the latter involves a single many-body Green's function, the former uses a multichannel Green's function in which two or more many-body Green's functions are coupled. Quasiparticles and satellites are thus naturally treated on equal footing in the multichannel Dyson equation. To assess the accuracy of our approach, we apply it here to the ground- and excited-state properties of the extended Hubbard dimer, an exactly solvable model for H2. In particular, we focus on the potential energy surface as well as the corresponding spectral functions and HOMO-LUMO gaps, which are well-known challenges for many-body approximations such as second Born and GW. We show that the multichannel Dyson equation gives overall very good results for all properties considered and outperforms both GW and second Born. In particular, the multichannel Dyson equation yields the correct ground-state energy and HOMO-LUMO gap in the dissociation limit contrary to GW.

DOI
10.1063/5.0291280

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We have recently presented the multichannel Dyson equation as an alternative to the standard single-channel Dyson equation. While the latter involves a single many-body Green's function, the former uses a multichannel Green's function in which two or more many-body Green's functions are coupled. Quasiparticles and satellites are thus naturally treated on equal footing in the multichannel Dyson equation. To assess the accuracy of our approach, we apply it here to the ground- and excited-state properties of the extended Hubbard dimer, an exactly solvable model for H2. In particular, we focus on the potential energy surface as well as the corresponding spectral functions and HOMO-LUMO gaps, which are well-known challenges for many-body approximations such as second Born and GW. We show that the multichannel Dyson equation gives overall very good results for all properties considered and outperforms both GW and second Born. In particular, the multichannel Dyson equation yields the correct ground-state energy and HOMO-LUMO gap in the dissociation limit contrary to GW.
We present the multichannel Dyson equation that combines two or more many-body Green's functions to describe the electronic structure of materials. In this work we use it to model photoemission spectra by coupling the one-body Green's function with the three-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 multichannel 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.
Recently we proposed a multichannel Dyson equation, in which we coupled the one- and the three-body Green's functions, to model photoemission spectra. In this work we extend this concept to higher-order Green's functions and to other spectroscopies. We show the general structure of the equations and how one can systematically approximate the corresponding multichannel self-energy. As a particular case, we focus on the coupling of the two-body and the four-body Green's functions in the electron-hole channel to describe neutral excitations. This formulation allows for the description of important many-body effects, such as biexcitons, in a natural way. We illustrate our approach by applying it to a two-level model system, which, in a one-particle picture, exhibits single and double excitations. Our method can correctly describe both kinds of excitation, unlike standard approaches, and in good agreement with the exact results.
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 work we use it to model photoemission spectra by coupling the one-body Green's function with the three-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.
We present an original strategy for the calculation of direct and inverse photo-emission spectra from first principles. The main goal is to go beyond the standard Green's function approaches, such as the GWGWGW 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. To this end we use as a key quantity the three-body Green's function, or, more precisely, its hole-hole-electron and electron-electron-hole parts, and we show how the one-body Green's function, and hence the corresponding spectral function, can be retrieved from it. We show that, contrary to the one-body Green's function, information about satellites is already present in the non-interacting three-body 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 give a proof of principle of our strategy by applying it to the Hubbard dimer, for which the exact self-energy is available.
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In a recent letter [Riva et al., Phys. Rev. Lett. 131, 216401 (2023)] 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, and (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.