Non-hermitian Green's function theory with $N$-body interactions: the coupled-cluster similarity transformation

Technical Summary: Non-hermitian Green's function theory with N-body interactions: the coupled-cluster similarity transformation

Problem Statement
The Green's function formalism is a cornerstone of ab initio many-body theory, providing a unified description of ground and excited states via the single-particle Green's function and the irreducible self-energy. Traditionally, this framework has been developed for hermitian Hamiltonians involving two-body (and recently three-body) interactions. Conversely, Coupled-Cluster (CC) theory is a primary method for obtaining ground-state properties, formulated via a non-hermitian similarity transformation of the Hamiltonian ($\bar{H} = e^{-T}He^T$). While previous attempts have combined these theories, they largely focused on constructing the electronic Green's function directly from CC eigenstates (IP/EA-EOM-CC) or connecting approximate self-energies (like $G_0W_0$) to CC. A rigorous, diagrammatic Green's function theory for the non-hermitian, $N$-body interaction Hamiltonian generated by the CC similarity transformation has remained unexplored. Specifically, the relationship between the functional-diagrammatic framework of Green's function theory and the CC similarity transformation requires a new formalism to handle the effective interactions arising from normal-ordering a non-hermitian $N$-body operator.

Methodology
The authors develop a novel Green's function formalism for general non-hermitian $N$-body interactions, specifically applied to the coupled-cluster similarity-transformed Hamiltonian. The methodology proceeds through several key theoretical steps:

1. Biorthogonal Quantum Theory: The authors extend standard quantum mechanical pictures (Schrödinger, Heisenberg, Interaction) to non-hermitian systems using a biorthogonal basis. They define left ($\langle \tilde{\Psi} |$) and right ($| \Psi \rangle$) eigenstates and establish a biorthogonal Interaction picture. This allows for the extension of the Gell-Mann and Low (GML) theorem to non-hermitian interactions, enabling the perturbative construction of correlation functions.
2. Definition of the Single-Particle Coupled-Cluster Green's Function (SP-CCGF): Unlike previous approaches that define the Green's function via similarity-transformed operators, this work defines the SP-CCGF ($\tilde{G}$) directly in the biorthogonal Heisenberg picture governed by the similarity-transformed Hamiltonian $\bar{H}$:
   $$i\tilde{G}_{pq}(t_1, t_2) = \langle \tilde{\Psi}_0 | T \{ a_p(t_1) a^\dagger_q(t_2) \} | \Phi_0 \rangle$$
   where $|\Phi_0\rangle$ is the reference determinant and $|\tilde{\Psi}_0\rangle$ is the left eigenstate.
3. Effective Interactions and Normal-Ordering: The authors demonstrate that constructing the self-energy as a functional of the exact Green's function ($\tilde{\Sigma}[\tilde{G}]$) requires normal-ordering the non-hermitian Hamiltonian with respect to the biorthogonal ground state. This generates effective one-body, two-body, and higher-body interactions ($\tilde{F}, \tilde{\Xi}, \tilde{\chi}$, etc.) that are functionals of $\tilde{G}$.
4. Diagrammatic Expansions:
   • Perturbative Expansion: The authors derive the perturbative expansion of the irreducible coupled-cluster self-energy ($\tilde{\Sigma}[G_0]$) with respect to the non-interacting reference Green's function ($G_0$). This expansion includes diagrams up to third order, revealing the emergence of effective interactions that vanish or simplify due to CC amplitude equations (e.g., the vanishing of the virtual-occupied block).
   • Self-Consistent Renormalization: Using the exact equation-of-motion for $\tilde{G}$, the authors derive the fully renormalized self-energy functional $\tilde{\Sigma}[\tilde{G}]$. This involves coupling the single-particle Green's function to higher-order Green's functions (4-point, 6-point, etc.) and defining corresponding vertex functions.
5. Connection to Lagrangian Derivatives: The work establishes a rigorous link between the diagrammatic expansion of the self-energy and the functional derivatives of the Coupled-Cluster Lagrangian with respect to the non-interacting Green's function.

Key Contributions and Results

• Novel Formalism: The paper presents the first diagrammatic theory of the irreducible self-energy and Bethe-Salpeter (BSE) kernel for a general non-hermitian $N$-body interaction, specifically tailored to the CC similarity transformation.
• Exact Dyson Equation: The authors derive an exact Dyson equation for the SP-CCGF, $\tilde{G} = G_0 + G_0 \tilde{\Sigma}[\tilde{G}] \tilde{G}$, where the self-energy is a functional of the exact coupled-cluster Green's function.
• Diagrammatic Consistency: The study details the perturbative expansion of $\tilde{\Sigma}$ up to third order. It shows that while the structure resembles standard electronic self-energies, the interaction vertices are replaced by effective interactions ($\tilde{\Xi}, \tilde{\chi}$) that depend on the Green's function. Crucially, diagrams involving four or more lines below a vertex vanish due to the structure of the CC similarity transformation.
• Static Component and Ground State Energy: The authors derive the exact static component of the coupled-cluster self-energy ($\tilde{\Sigma}^\infty$) and demonstrate that the coupled-cluster ground state energy can be recovered from the self-energy and the associated density matrix, resolving previous discrepancies in energy definitions between CC and Green's function approaches.
• Bethe-Salpeter Kernel: The paper derives the diagrammatic expansion for the coupled-cluster BSE kernel ($\tilde{\Xi} = i \delta \tilde{\Sigma} / \delta \tilde{G}$), highlighting the complexity introduced by the functional dependence of interaction lines on $\tilde{G}$.
• CC-$G_0W_0$ Approximation: By leveraging the connections between the Green's function formalism and CC theory, the authors introduce a "CC-$G_0W_0$" self-energy. This approximation utilizes the ring-CCD (rCCD) approximation and truncates the interaction matrices to 2-particle-1-hole/2-hole-1-particle (2p1h/2h1p) excitation spaces, providing a bridge between GW theory and CC theory.
• Dyson Supermatrix: The spectral representation of the self-energy is used to construct a coupled-cluster Dyson supermatrix, which yields the exact ionization potentials and electron affinities, analogous to the electronic Dyson supermatrix but adapted for the non-hermitian CC context.

Significance
The paper claims to provide a "rigorous formulation and extension" of previous results (specifically Ref. [43]) to coupled-cluster theory formulated about an arbitrary reference state. By unifying the techniques of CC and Green's function theory, the work offers a "formally exact" framework that clarifies the nature of the many-body self-energy and Bethe-Salpeter kernel in the context of wavefunction-based approaches. The authors emphasize that this change in perspective—treating the similarity-transformed Hamiltonian as the fundamental interaction—unveils the derivation of the self-energy and Green's function that naturally arise within CC theory. This formalism is presented as a necessary step to fully understand the functional-diagrammatic relationships in non-hermitian many-body systems and to develop new approximations (like CC-$G_0W_0$) that combine the strengths of both theoretical frameworks. The work does not propose new experimental applications but rather establishes the theoretical groundwork for future computational developments in condensed matter and nuclear physics.