The algebraic diagrammatic construction scheme at nth order includes all Feynman diagrams up to nth order and a partial summation of diagrams arising from the latter to infinite order. In the present work this scheme is carried out for the first time at third order for the particle-particle propagator. Because of the many diagrams involved in third order the approach requires a very careful analysis of terms which describe the corresponding perturbation expansion of the propagator. The relevant technical details of the analysis are presented.
The algebraic diagrammatic construction scheme at nth order includes all Feynman diagrams up to nth order and a partial summation of diagrams arising from the latter to infinite order. In the present work this scheme is carried out for the first time at third order for the particle-particle propagator. Because of the many diagrams involved in third order the approach requires a very careful analysis of terms which describe the corresponding perturbation expansion of the propagator. The relevant technical details of the analysis are presented.
Background: Self-consistent Green's function theory has recently been extended to the basic formalism needed to account for three-body interactions [Carbone, Cipollone, Barbieri, Rios, and Polls, Phys. Rev. C 88, 054326 (2013)]. The contribution of three-nucleon forces has so far been included in ab initio calculations on nuclear matter and finite nuclei only as averaged two-nucleon forces.Purpose: We derive the working equations for all possible two- and three-nucleon terms that enter the expansion of the self-energy up to the third order, thus including the interaction-irreducible (i.e., not averaged) diagrams with three-nucleon forces that have been previously neglected.Methods: We employ the algebraic diagrammatic construction up to the third order as an organization scheme for generating a nonperturbative self-energy, in which ring (particle-hole) and ladder (particle-particle) diagrams are resummed to all orders.Results: We derive expressions of the static and dynamic self-energy up to the third order, by taking into account the set of diagrams required when either the skeleton or nonskeleton expansions of the single-particle propagator are assumed. A hierarchy of importance among different diagrams is revealed, and a particular emphasis is given to a third-order diagram [see Fig. 2(c)] that is expected to play a significant role among those featuring an interaction-irreducible three-nucleon force.Conclusion: A consistent formalism to resum at infinite order correlations induced by three-nucleon forces in the self-consistent Green's function theory is now available and ready to be implemented in the many-body solvers.
A purely algebraic approach to Green's functions is presented. The new formulation allows the determination of a closed-form equation for Green's functions which depends only on the ground state of the system. By a straightforward perturbation expansion of this state an interesting approximation scheme is obtained which is nonperturbative in the final states (e.g., excited or ionic states). This method, which can be used for various propagators, is explicitly applied to the particle-particle propagator carrying out the scheme up to third order. The approach is discussed in detail and briefly compared with the well-established algebraic diagrammatic construction approximation scheme.
Background: The Gorkov approach to self-consistent Green's function theory has been formulated by Som\`a, Duguet, and Barbieri in [Phys. Rev. C 84, 064317 (2011)]. Over the past decade, it has become a method of reference for first-principles computations of semimagic nuclear isotopes. The currently available implementation is limited to a second-order self-energy and neglects particle-number nonconserving terms arising from contracting three-particle forces with anomalous propagators. For nuclear physics applications, this is sufficient to address first-order energy differences (i.e., two neutron separation energies, excitation energies of states dominating the one-nucleon spectral function), ground-state radii and moments on an accurate enough basis. However, addressing absolute binding energies, fine spectroscopic details of $N\ifmmode\pm\else\textpm\fi{}1$ particle systems or delicate quantities such as second-order energy differences associated with pairing gaps, requires going to higher truncation orders.Purpose: The formalism is extended to third order in the algebraic diagrammatic construction (ADC) expansion with two-body Hamiltonians.Methods: The expansion of Gorkov propagators in Feynman diagrams is combined with the algebraic diagrammatic construction up to the third order as an organization scheme to generate the Gorkov self-energy.Results: Algebraic expressions for the static and dynamic contributions to the self-energy, along with equations for the matrix elements of the Gorkov eigenvalue problem, are derived. It is first done for a general basis before specifying the set of equations to the case of spherical systems displaying rotational symmetry. Workable approximations to the full self-consistency problem are also elaborated on. The formalism at third order it thus complete for a general two-body Hamiltonian.Conclusions: Working equations for the full Gorkov-ADC(3) are now available for numerical implementation.
A new approach to the one-particle Green's functions G for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part M related to G via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematical way a set of approximation schemes (nth-order ADC schemes) that represent infinite partial summations for M and (via the Dyson equation) for G being complete through nth order of perturbation theory. The corresponding mathematical procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed. While the second- and third-order schemes can be viewed as systematic rederivations of previous approximation schemes, the fourth-order ADC scheme represents a complete fourth-order approximation for the self-energy and the one-particle Green's function which was hitherto not available.
Using a general-order ab initio many-body Green's function method, we numerically illustrate several pathological behaviors of the Feynman-Dyson diagrammatic perturbation expansion of one-particle many-body Green's functions as electron Feynman propagators. (i) The perturbation expansion of the frequency-dependent self-energy is not convergent at the exact self-energy in many frequency domains. (ii) An odd-perturbation-order self-energy has a qualitatively wrong shape and, as a result, many roots of the corresponding Dyson equation are nonphysical in that the poles may be complex or residues can exceed unity or be negative. (iii) A higher even-order self-energy consists of vertical lines at many frequencies, predicting numerous phantom poles with zero residues. (iv) Infinite partial resummations of diagrams by vertex or edge renormalization tend to exacerbate these pathologies. (v) The nonconvergence is caused by the nonanalyticity of the rational-function form of the exact Green's function at many frequencies, where the radius of convergence of its Taylor expansion is zero. This is consistent with the fact that (vi) Pad\'e approximants (power-series expansions of a rational function) can largely restore the correct shape and poles of the Green's function. Nevertheless, not only does the nonconvergence render higher-order Feynman-Dyson diagrammatic perturbation theory useless for many lower-lying ionization or higher-lying electron-attachment states but it also calls into question the validity of its combined use with the Ans\"atze requiring the knowledge of all poles and residues. Such Ans\"atze include the Galitskii-Migdal identity, the self-consistent Green's function methods, and some models of the algebraic diagrammatic construction.
Within the framework of the many-body Green's-function method we present a new approach to the polarization propagator for finite Fermi systems. This approach makes explicit use of the diagrammatic perturbation expansion for the polarization propagator, and reformulates the exact summation in terms of a simple algebraic scheme, referred to as the algebraic diagrammatic construction (ADC). The ADC defines in a natural way a set of approximation schemes (nth-order ADC schemes) which represent infinite partial summations exact up to nth order of perturbation theory. In contrast to the random-phase-approximation (RPA)-like schemes, the corresponding mathematical procedures are essentially Hermitian eigenvalue problems in limited configuration spaces of unperturbed excited configurations. Explicit equations for the first- and second-order ADC schemes are derived. These schemes are thoroughly discussed and compared with the Tamm-Dancoff approximation and RPA schemes.