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arXiv:2503.06586 · Physical Review Research

Non-Hermitian Green’s function theory with N -body interactions: The coupled-cluster similarity transformation

Christopher J. N. Coveney, David P. Tew

Published Feb 17, 2026• 4 citations
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Abstract

We present the diagrammatic theory of the irreducible self-energy and Bethe-Salpeter kernel that naturally arises within the Green’s function formalism for a general N -body non-Hermitian interaction. In this work, we focus specifically on the coupled-cluster (CC) self-energy generated by the similarity transformation of the electronic structure Hamiltonian. The main result of this work is a Green’s function formalism for non-Hermitian many-body quantum systems. We develop the biorthogonal quantum theory to construct dynamical correlation functions where the time dependence of operators is governed by a non-Hermitian Hamiltonian. We extend the Gell-Mann and Low theorem to include non-Hermitian interactions and to generate perturbative expansions of many-body Green’s functions. We introduce the single-particle coupled-cluster Green’s function and derive the perturbative diagrammatic expansion for the non-Hermitian coupled-cluster self-energy in terms of the “noninteracting” reference Green’s function, Σ ̃ [ G 0 ] . From the exact equation of motion of the single-particle coupled-cluster Green’s function, we derive the self-consistent renormalized coupled-cluster self-energy, Σ ̃ [ G ̃ ] , and demonstrate its relationship to the perturbative expansion of the self-energy, Σ ̃ [ G 0 ] . Subsequently, we show that the usual electronic self-energy can be recovered from the coupled-cluster self-energy by neglecting the effects of the similarity transformation. We show how the coupled-cluster ground-state energy can be obtained from the coupled-cluster self-energy and provide an overview of the relationship between approximations for the coupled-cluster self-energy, ionization potential/electron affinity equation-of-motion coupled-cluster theory, and the G 0 W 0 approximation. As a result, we introduce the CC- G 0 W 0 self-energy by leveraging the connections between Green’s function and coupled-cluster theory. Finally, we derive the diagrammatic expansion of the coupled-cluster Bethe-Salpeter kernel.

DOI
10.1103/9vdr-vhxg

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We present the diagrammatic theory of the irreducible self-energy and Bethe-Salpeter kernel that naturally arises within the Green’s function formalism for a general N -body non-Hermitian interaction. In this work, we focus specifically on the coupled-cluster (CC) self-energy generated by the similarity transformation of the electronic structure Hamiltonian. The main result of this work is a Green’s function formalism for non-Hermitian many-body quantum systems. We develop the biorthogonal quantum theory to construct dynamical correlation functions where the time dependence of operators is governed by a non-Hermitian Hamiltonian. We extend the Gell-Mann and Low theorem to include non-Hermitian interactions and to generate perturbative expansions of many-body Green’s functions. We introduce the single-particle coupled-cluster Green’s function and derive the perturbative diagrammatic expansion for the non-Hermitian coupled-cluster self-energy in terms of the “noninteracting” reference Green’s function, Σ ̃ [ G 0 ] . From the exact equation of motion of the single-particle coupled-cluster Green’s function, we derive the self-consistent renormalized coupled-cluster self-energy, Σ ̃ [ G ̃ ] , and demonstrate its relationship to the perturbative expansion of the self-energy, Σ ̃ [ G 0 ] . Subsequently, we show that the usual electronic self-energy can be recovered from the coupled-cluster self-energy by neglecting the effects of the similarity transformation. We show how the coupled-cluster ground-state energy can be obtained from the coupled-cluster self-energy and provide an overview of the relationship between approximations for the coupled-cluster self-energy, ionization potential/electron affinity equation-of-motion coupled-cluster theory, and the G 0 W 0 approximation. As a result, we introduce the CC- G 0 W 0 self-energy by leveraging the connections between Green’s function and coupled-cluster theory. Finally, we derive the diagrammatic expansion of the coupled-cluster Bethe-Salpeter kernel.
Coupled-cluster and Green's function theories are highly successful in treating many-body electron correlation, and there has been significant interest in identifying and leveraging connections between them. Here we present a diagrammatic definition of the irreducible coupled-cluster self-energy that directly embeds coupled-cluster (CC) theory within the framework of many-body field theory. The equation-of-motion coupled-cluster (EOM-CC) treatment emerges naturally from our definition via the Dyson equation and the Bethe-Salpeter equation (BSE), providing a unified description of the random phase approximation (RPA), G W -BSE, and CC theory for ground state and excitation energies. This clarifies the origin of previously established connections between RPA, G W -BSE, and CC theory, and it exposes the relationship between vertex corrections and the coupled-cluster amplitude equations.
Connection between GWGWGW and Extended Coupled Cluster
Coupled-cluster (CC) theory and Green's function many-body perturbation theory (MBPT) have long evolved as distinct yet complementary frameworks for describing electronic correlation. While CC methods employ exponential wavefunction parametrizations that guarantee size extensivity and systematic improvability, Green's function approaches such as the GWGWGW approximation describe quasiparticle and optical excitations through diagrammatic resummations. Recent analyses have established a formal correspondence between these frameworks: the GWGWGW approximation is equivalent to an equation-of-motion (EOM) treatment of the direct-ring coupled-cluster doubles (drCCD) method. Within this context, the extended CC (ECC) ansatz offers a unified framework connecting CC and MBPT. This formulation bridges CC-based and Green's function-based methods, providing novel avenues for incorporating vertex corrections within a CC framework that preserves a sum-over-state representation of the self-energy and lead to potentially systematically improvable Green's function approaches.
Coupled-cluster (CC) theory and Green’s function many-body perturbation theory (MBPT) have long evolved as distinct yet complementary frameworks for describing electronic correlation. While CC methods employ exponential wave function parametrizations that guarantee size extensivity and systematic improvability, Green’s function approaches, such as the GW approximation, describe quasiparticle and optical excitations through diagrammatic resummations. Recent analyses have established a formal correspondence between these frameworks: the GW approximation is equivalent to an equation-of-motion (EOM) treatment of the direct-ring coupled-cluster doubles (drCCD) method. Within this context, the extended CC (ECC) ansatz offers a unified framework connecting CC and MBPT. This formulation bridges CC-based and Green’s function-based methods, providing novel avenues for incorporating vertex corrections within a CC framework that preserves a sum-over-state representation of the self-energy and leads to potentially systematically improvable Green’s function approaches.
Here, we build on the works of Scuseria et al. [J. Chem. Phys. 129, 231101 (2008)] and Berkelbach [J. Chem. Phys. 149, 041103 (2018)] to show connections between the Bethe–Salpeter equation (BSE) formalism combined with the GW approximation from many-body perturbation theory and coupled-cluster (CC) theory at the ground- and excited-state levels. In particular, we show how to recast the GW and Bethe–Salpeter equations as non-linear CC-like equations. Similitudes between BSE@GW and the similarity-transformed equation-of-motion CC method are also put forward. The present work allows us to easily transfer key developments and the general knowledge gathered in CC theory to many-body perturbation theory. In particular, it may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the GW and BSE frameworks.
The quantum theory of many-particle systems is deeply connected to the mathematical formalism and techniques of quantum field theory, which are crucial in order describe the emergent phenomena that arise in condensed matter physics. The main goal of this thesis is to provide a unified description of quantum field theory methods in the context of many-particle physics in order to extend and develop these techniques to better understand the different quantum properties of matter. This both furthers the conceptual understanding of quantum many-body correlation and also allows for the development of novel theoretical techniques that combine the different formalisms employed in condensed matter theory, quantum chemistry and nuclear physics. To achieve this aim, the thesis is composed of two parts. Part I deals with many-body fermion-fermion interactions and is physically motivated by the non-relativistic electronic structure problem. Here, we introduce the Keldysh contour formulation of quantum field theory to provide a unified presentation of quantum systems both in and out-ofequilibrium. Then, we extend the conventional quantum theory of many-particle systems, based on hermitian quantum mechanics, to describe systems governed by non-hermitan Hamiltonians. This allows us to present a natural unification of two major pillars in the ab initio quantum theory of many-particle systems: coupled-cluster theory and the Green’s function formalism. This unification is further demonstrated through the emergence of the coupled-cluster amplitude equations from the Green’s function formalism. In particular, this analysis has direct bearing on the development of novel theoretical approaches to the electronic structure and nuclear many-body problems. Our findings are also relevant to the study of general non-hermitian as well as open quantum systems. Part II focuses on the quantum field theory of electron-nuclear interactions in infinite solids. Here, we use the Keldysh contour representation presented in Part I to understand excited state dynamics that include electron-electron and electron-lattice interactions in materials. We also derive the analytic continuation of the Green’s function for systems that have continuous eigenspectra. This analysis demonstrates the relationship between the Green’s function formalism and scattering theory, making the role of the continuum and causality conditions explicitly clear in the context of the pole condition of the Dyson equation. Using the analytic continuation of the propagator, we provide an in-depth analysis of electron-phonon and exciton-phonon scattering channels, with a particular focus on phonon-driven exciton dissociation scattering. Our theoretical analysis then firmly demonstrates the connection between the extension of the Bethe-Salpeter equation approach to include electron-phonon interactions and effective field theories based on composite exciton-phonon coupling. We subsequently combine the techniques presented in Part I, based on the Dyson supermatrix formulation, with the analysis presented in Part II to calculate, from first principles, the effects of including electron-phonon interactions on the excited states of heterogeneous semiconductors. Finally, we provide a summary of the fundamental contributions presented in this thesis and the potential further research directions to be pursued.
In this paper we discuss a new formalism for producing an analytic coupled-cluster (CC) Green's function for an N-electron system by shifting the poles of similarity transformed Hamiltonians represented in N - 1 and N + 1 electron Hilbert spaces. Simple criteria are derived for the states in N - 1 and N + 1 electron spaces that are then corrected in the spectral resolution of the corresponding matrix representations of the similarity transformed Hamiltonian. The accurate description of excited state processes within a Green's function formalism would be of significant importance to a number of scientific communities ranging from physics and chemistry to engineering and the biological sciences. This is because the Green's function methodology provides a direct path for not only calculating properties whose underlying origins come from coupled many-body interactions but also provides a straightforward path for calculating electron transport, response, and correlation functions that allows for a direct link with experiment. As a special case of this general formulation, we discuss the application of this technique for Green's function defined by the CC with singles and doubles representation of the ground-state wave function.