We propose a novel many-body truncation for Gorkov self-consistent Green's function (SCGF) theory where pairing correlations are handled at first order, while dynamical correlations are described using the particle-number-conserving Dyson-SCGF scheme up to third order in the algebraic diagrammatic construction. The new method is enabled by the introduction of a scheme that allows to approximate the Gorkov propagator in terms of a particle-number-conserving optimized reference state. The approach provides state-of-the-art predictions of the equation of state and spectral properties of infinite nuclear matter at zero temperature and in the presence of pairing. We find satisfactory results using modern saturating Hamiltonians at next-to-next-to-leading order in chiral effective field theory.
We propose a novel many-body truncation for Gorkov self-consistent Green's function (SCGF) theory where pairing correlations are handled at first order, while dynamical correlations are described using the particle-number-conserving Dyson-SCGF scheme up to third order in the algebraic diagrammatic construction. The new method is enabled by the introduction of a scheme that allows to approximate the Gorkov propagator in terms of a particle-number-conserving optimized reference state. The approach provides state-of-the-art predictions of the equation of state and spectral properties of infinite nuclear matter at zero temperature and in the presence of pairing. We find satisfactory results using modern saturating Hamiltonians at next-to-next-to-leading order in chiral effective field theory.
We propose a novel many-body truncation for Gorkov self-consistent Green's function (SCGF) theory where pairing correlations are handled at first order, while dynamical correlations are described using the particle-number-conserving Dyson-SCGF scheme up to third order in the algebraic diagrammatic construction.The new method is enabled by the introduction of a scheme that allows to approximate the Gorkov propagator in terms of a particle-number-conserving optimized reference state.The approach provides state-of-the-art predictions of the equation of state and spectral properties of infinite nuclear matter at zero temperature and in the presence of pairing.We find satisfactory results using modern saturating Hamiltonians at next-to-next-to-leading order in chiral effective field theory.
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.
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 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.
The multichannel Dyson equation (MCDE) was recently introduced as a new approximation scheme to compute the one-body Green function in many-body systems, as reported by Riva et al. in Physical Review Letters, volume 131, article 216401, published in 2023. The physical content of this novel approximation scheme is further clarified by recovering it from an extended version of the algebraic diagrammatic construction (ADC) truncation scheme. It is thus demonstrated that the MCDE approximation lies in between the so-called ADC(2) and ADC(3) truncations of the dynamical self energy. Building on this clarification, the MCDE approximation is tested on the periodic one-dimensional Hubbard model with 4, 6, and 8 site lattices and shown to deliver an improved treatment over ADC(2) of both the quasiparticle peaks and the so-called satellites in the spectral strength distribution.
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.