The GW approximation is a cornerstone of many-body perturbation theory for computing single-particle excitations, yet it can fail qualitatively in strongly correlated systems where the single-reference picture becomes inadequate. To overcome this limitation, we develop a diagrammatic route to multi-reference GW (MR-GW) from an interacting active-space reference, treating strong correlation nonperturbatively and the residual interaction through dynamical screening at the multi-reference random phase approximation level. Because the conventional diagrammatic self-energy expansion no longer applies, we define a practical one-shot MR-GW approximation within a generalized diagrammatic framework, recovering standard GW as a special case. Applications to prototypical correlated atoms and molecules show that the interacting active-space reference captures multiconfigurational satellite structures poorly described by single-reference GW, while dynamical screening generally improves the satellite energies as well as the leading ionization and electron-attachment energies. This work opens a diagrammatic route to Green's-function methods that combine nonperturbative strong correlation with dynamical screening.
The GW approximation is a cornerstone of many-body perturbation theory for computing single-particle excitations, yet it can fail qualitatively in strongly correlated systems where the single-reference picture becomes inadequate. To overcome this limitation, we develop a diagrammatic route to multi-reference GW (MR-GW) from an interacting active-space reference, treating strong correlation nonperturbatively and the residual interaction through dynamical screening at the multi-reference random phase approximation level. Because the conventional diagrammatic self-energy expansion no longer applies, we define a practical one-shot MR-GW approximation within a generalized diagrammatic framework, recovering standard GW as a special case. Applications to prototypical correlated atoms and molecules show that the interacting active-space reference captures multiconfigurational satellite structures poorly described by single-reference GW, while dynamical screening generally improves the satellite energies as well as the leading ionization and electron-attachment energies. This work opens a diagrammatic route to Green's-function methods that combine nonperturbative strong correlation with dynamical screening.
Many-body perturbation theory (MBPT) based on Green’s functions and Feynman diagrams provides a fundamental theoretical framework for various ab initio computational approaches in molecular and materials science, including the random phase approximation (RPA) and GW approximation. Unfortunately, this perturbation expansion often fails in systems with strong multireference characters. Extending diagrammatic MBPT to the multireference case is highly nontrivial and remains largely unexplored, primarily due to the breakdown of Wick’s theorem. In this work, we develop a diagrammatic multireference generalization of MBPT for computing correlation energies of strongly correlated systems, by using the cumulant expansion of many-body Green’s functions in place of Wick’s theorem. This theoretical framework bridges the gap between MBPT in condensed matter physics and multireference perturbation theories (MRPT) in quantum chemistry, which had been almost exclusively formulated within time-independent wave function frameworks prior to this work. Our formulation enables the explicit incorporation of strong correlation effects from the outset as in MRPT, while treating residual weak interactions through a generalized diagrammatic perturbation expansion as in MBPT. As a concrete demonstration, we formulate a multireference (MR) extension of the standard single-reference (SR) RPA by systematically resumming generalized ring diagrams, which naturally leads to a unified set of equations applicable to both SR and MR cases. Benchmark calculations on prototypical molecular systems reveal that MR-RPA successfully resolves the well-known failure of SR-RPA in strongly correlated systems. This theoretical advancement paves the way for advancing ab initio computational methods through diagrammatic resummation techniques in future.
A diagrammatic multi-reference generalization of many-body perturbation theory was recently introduced [Wang et al., J. Phys. Chem. Lett. 16, 3047 (2025)]. This framework allows us to extend single-reference (SR) Green's function methods defined at the diagrammatic level naturally to the multi-reference case, as previously exemplified by the formulation of multi-reference direct random phase approximation (MR-dRPA) and the multi-reference second-order screened exchange approximation. In this work, we further elaborate on this framework and use it to develop MR generalizations of two other RPA variants, namely, particle-hole (ph) RPA with exchange (MR-RPAx) and particle-particle RPA (MR-ppRPA). We define these two MR generalizations by infinite-order resummations of the generalized "ring" and "ladder" diagrams with antisymmetrized interaction vertices, respectively, which incorporate the contributions from the active-space connected two-body Green's functions. As for MR-dRPA, we derive unified sets of equations that hold at both SR and MR levels for RPAx and ppRPA, respectively. We perform numerical studies of prototypical systems using the three MR-RPA methods and carry out a perturbative analysis to gain a deeper understanding of their behaviors. We find that error cancellation between the second and third orders is a key factor for both SR-RPA and MR-RPA. In addition, we observe that MR-phRPA (MR-dRPA and MR-RPAx) and MR-ppRPA tend to overestimate and underestimate correlation energies, respectively, suggesting that better accuracy can be achieved by further combining these two channels in the future.
In order to properly understand the utility of many-body perturbation theory as applied to finite systems, we use the configuration interaction approach to compute the electronic self-energy. The validity of the commonly used GW approximation from many-body perturbation theory is tested by comparing the self-energy explicitly computed according to the diagrammatic expansions to that from the inversion of the Dyson equation. It is constructed as a functional of the interacting Green function G and screened Coulomb interaction W. W is explicitly computed through the diagonalization of the many-body Hamiltonian and, thus, takes polarization effects into account beyond the random phase approximation. The Na9+ cluster and the dissociation of the C2 molecule are discussed as examples. We find that the GW approximation yields accurate results for weakly correlated systems with a predominantly single-determinant ground state such as Na9+ clusters. The C2 molecule is a pathological system with multideterminantal ground state where the GW approach ceases to be valid.
For the paradigmatic case of H(2) dissociation, we compare state-of-the-art many-body perturbation theory in the GW approximation and density-functional theory in the exact-exchange plus random-phase approximation (RPA) for the correlation energy. For an unbiased comparison and to prevent spurious starting point effects, both approaches are iterated to full self-consistency (i.e., sc-RPA and sc-GW). The exchange-correlation diagrams in both approaches are topologically identical, but in sc-RPA they are evaluated with noninteracting and in sc-GW with interacting Green functions. This has a profound consequence for the dissociation region, where sc-RPA is superior to sc-GW. We argue that for a given diagrammatic expansion, sc-RPA outperforms sc-GW when it comes to bond breaking. We attribute this to the difference in the correlation energy rather than the treatment of the kinetic energy.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
Strongly correlated quantum field theories present a formidable challenge to conventional theoretical methods, particularly those rooted in perturbative expansions. The breakdown of perturbation theory in regimes where interaction strengths are comparable to or exceed kinetic energies necessitates the development of non-perturbative approaches. This paper introduces and elaborates on a novel framework for diagrammatic renormalization specifically designed for strongly correlated quantum fields, moving beyond the limitations inherent in iterative or weak-coupling expansions. We explore how a systematic resummation of infinite classes of diagrams, guided by a functional renormalization group perspective, can capture the complex interplay of quantum fluctuations and emergent phenomena characteristic of these systems. Our approach leverages a self-consistent treatment of Green's functions and self-energies, allowing for the faithful description of spectral functions, phase transitions, and exotic ground states in a manner that traditional perturbation theory cannot. We discuss the theoretical underpinnings, computational strategies, and potential applications across diverse domains, including condensed matter physics, quantum chromodynamics, and ultracold atomic systems. The methodology aims to provide a robust, diagrammatically transparent pathway to understanding the non-trivial fixed points and scaling behaviors that govern strongly correlated phenomena, offering a powerful alternative where standard approximations fail.