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.
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.
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.
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.
Scope The book focuses primarily on many-body (or better, many-electron) methods for electron correlation. These include Rayleigh–Schrödinger perturbation theory (RSPT) , particularly in its diagrammatic representation (referred to as many-body perturbation theory , or MBPT ), and coupled-cluster (CC) theory; their relationship to configuration interaction (CI) is included. Further extensions address properties other than the energy, and also excited states and multireference CC and MBPT methods. The many-body algebraic and diagrammatic methods used in electronic structure theory have their origin in quantum field theory and in the study of nuclear matter and nuclear structure. The second-quantization formalism was first introduced in a treatment of quantized fields by Dirac (1927) and was extended to fermion systems by Jordan and Klein (1927) and by Jordan and Wigner (1928). This formalism is particularly useful in field theory, in scattering problems and in the study of infinite systems because it easily handles problems involving infinite, indefinite or variable numbers of particles. The diagrammatic approach was introduced into field theory by Feynman (1949a,b) and applied to many-body systems by Hugenholtz (1957) and by Goldstone (1957). Many-body perturbation theory and its linked-diagram formalism were first introduced by Brueckner and Levinson (1955) and by Brueckner (1955), and were formalized by Goldstone (1957). Other important contributions to the methodology, first in field theory and then in the theory of nuclear structure, are due to Dyson (1949a,b), Wick (1950), Hubbard (1957, 1958a,b) and Frantz and Mills (1960). Applications to the electronic structure of atoms and molecules began with the work of Kelly (1963, 1964a,b, 1968), and molecular applications using finite analytical basis sets appeared in the work of Bartlett and Silver (1974a, b).
Accurately describing many-body effects in multi-orbital systems remains a major challenge in theoretical condensed matter physics. At present, there is a significant methodological gap between the numerical tools used in ab initio computational materials science and those developed to study strong electronic correlations. The former can treat realistic, large-scale systems but typically neglect many-body effects, while the latter focus on simplified models with only a few degrees of freedom, as only such models can be solved accurately in the presence of strong interactions. The purpose of this thesis is to bridge these two approaches and establish a systematic theoretical framework for realistic correlated electronic materials. This involves a full-cycle methodology that begins with constructing ab initio interacting models from density-functional theory, solving them using dynamical mean-field theory to capture local correlations, and extending beyond to incorporate non-local collective electronic fluctuations. To this end, we introduce the "dual" approach to strong correlations, which includes the dual fermion, dual boson, and dual triply irreducible local expansion methods. The central idea of the dual theories is to shift the reference point of the conventional Feynman diagrammatic expansion from a non-interacting electronic system to an interacting but exactly solvable one. Integrating out this reference system recasts the expansion in an effective dual space, where all diagrammatic building blocks are renormalized by the corresponding impurity quantities. This procedure transforms a non-perturbative expansion in the original variables into a perturbative expansion in terms of dual fermionic and bosonic fields, exact in both the weak- and strong-coupling limits. In this thesis, we collect and systematize the major developments of dual techniques achieved to date.
Many-body perturbation theory (MBPT) is widely used in quantum physics, chemistry, and materials science. At the heart of MBPT is the Feynman diagrammatic expansion, which is, simply speaking, an elegant way of organizing the combinatorially growing number of terms of a certain Taylor expansion. In particular, the construction of the `bold Feynman diagrammatic expansion' involves the partial resummation to infinite order of possibly divergent series of diagrams. This procedure demands investigation from both the combinatorial (perturbative) and the analytical (non-perturbative) viewpoints. In Part I of this two-part series, we illustrate how the combinatorial properties of Feynman diagrams in MBPT can be studied in the simplified setting of Gibbs measures (known as the Euclidean lattice field theory in the physics literature) and provide a self-contained explanation of Feynman diagrams in this setting. We prove the combinatorial validity of the bold diagrammatic expansion, with methods generalizable to several types of field theories and interactions. Our treatment simplifies the presentation and numerical study of known techniques in MBPT such as the self-consistent Hartree-Fock approximation (HF), the second-order Green's function approximation (GF2), and the GW approximation. The bold diagrams are closely related to the Luttinger-Ward (LW) formalism, which was proposed in 1960 but whose analytic properties have not been rigorously established. The analytical study of the LW formalism in the setting of Gibbs measures will be the topic of Part II.