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.
In this paper some new physical notations are given for the Green's functions and equations of motion (EOM) in many body physics with the concept of quasiparticles. It shows how the many body correlations existing in many body systems can be taken into account through the derivation of EOMs self-consistently. The scheme is compared with the expansion of Feynman diagrams. Finally some further discussions are given for the unique properties of EOM techniques in helping to model a system and the interactions self-consistently.
On the basis of an exact perturbational expression for the interacting one-particle Green function
G corresponding to bosons / fermions in terms of the bare interaction potential
v and permanents / determinants of the non-interacting one-particle Green function
G0, we deduce four recursive perturbation expansions for the self-energy
Σ. With
W denoting the dynamic screened interaction potential, these perturbation expansions are identical to those of
Σ in terms of (i) self-energy diagrams and
(v,G0), (ii)
G-skeleton self-energy diagrams and
(v,G), (iii)
W-skeleton self-energy diagrams and
(W,G0), and (iv)
G- and
W-skeleton self-energy diagrams and
(W,G). For the calculation of
W, we rely on a similar exact perturbational expression for the interacting two-particle Green function
G2 as for
G. From this expression, we deduce four recursive perturbation expansions for the polarization function
P, necessary for the calculation of
W, that are similar to those for
Σ specified above. The correlation functions considered in this paper may be corresponding to ground states, and thermal ensemble of states. For thermal ensemble of states, we consider both the imaginary-time formalism of Matsubara, and the real-time formalism of thermo-field dynamics (TFD). The latter is advantageous for the direct calculation of dynamic correlation functions. In an appendix, we apply the formalisms presented in this paper to the Hubbard Hamiltonian for spin-
21 fermions on a lattice in arbitrary
d spatial dimensions. In two further appendices, we present methods and short programs for determining the
νth-order diagrams corresponding to the perturbation expansions of
G in terms of
(v,G0), and
Σ in terms of
(v,G0) and
(v,G) on the basis of the cycle decompositions of the elements of the symmetric group
S2ν.
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.
We report an implementation of self-consistent Green's function many-body theory within a second-order approximation (GF2) for application with molecular systems. This is done by iterative solution of the Dyson equation expressed in matrix form in an atomic orbital basis, where the Green's function and self-energy are built on the imaginary frequency and imaginary time domain, respectively, and fast Fourier transform is used to efficiently transform these quantities as needed. We apply this method to several archetypical examples of strong correlation, such as a H32 finite lattice that displays a highly multireference electronic ground state even at equilibrium lattice spacing. In all cases, GF2 gives a physically meaningful description of the metal to insulator transition in these systems, without resorting to spin-symmetry breaking. Our results show that self-consistent Green's function many-body theory offers a viable route to describing strong correlations while remaining within a computationally tractable single-particle formalism.
The method of correlated basis functions provides a means of systematically exploiting our intuitive understanding of real physical systems. Here we seek especially to develop a formalism for describing systems of strongly interacting fermions, for example, nuclei and liquid
He3, using a complete set of correlated functions ${\ensuremath{\Psi}}_{\mathrm{m}}=\frac{F{\ensuremath{\Phi}}_{\mathrm{m}}}{{({\ensuremath{\Phi}}_{\mathrm{m}}, {F}^{2}{\ensuremath{\Phi}}_{\mathrm{m}})}^{\frac{1}{2}}}$.
F is taken as a product of two-body factors, $\ensuremath{\Pi}{i<j}^{}[f({r}_{\mathrm{ij}})]$, the
f(rij) embodying at least the correlations arising from strong short-range repulsions; and ${{\ensuremath{\Phi}}_{\mathrm{m}}}$, a set of independent-particle-model functions which might serve as a basis in the absence of highly singular interactions. The matrix elements
Hmn,
Nmn of the Hamiltonian and identity operators with respect to the correlated basis ${{\ensuremath{\Psi}}_{\mathrm{m}}}$ are evaluated by generalization of cluster-expansion techniques developed originally for the calculation of the partition function of a classical imperfect gas and adapted to the quantum-mechanical many-body problem by Iwamoto and Yamada. Contributions to our expansions for the
Hmn are classified according to order of magnitude in the quantity $\ensuremath{\rho}\ensuremath{\omega}$, where $\ensuremath{\omega}=\ensuremath{\int}({f}^{2}(r)\ensuremath{-}1)d\mathrm{r}$ is a "correlation parameter" and $\ensuremath{\rho}$ the average particle density. To the extent that $|\ensuremath{\rho}\ensuremath{\omega}|$ is small, it provides a true expansion parameter. A detailed technical discussion of the cluster expansions is given, with special emphasis on their symmetry properties upon truncation. Although the many-body problem is soluble by simultaneous diagonalization of the matrices (
Hmn) and (
Nmn), it is preferable first to transform to an orthonormal basis of "unperturbed" correlated functions ${{\ensuremath{\Theta}}_{\mathrm{m}}}$ with respect to which the Hamiltonian and identity operators have the matrix representations (
Hmn) and $({N}_{\mathrm{mn}})=({\ensuremath{\delta}}_{\mathrm{mn}})$, and then to attempt the separate diagonalization of (
Hmn). In general, such a transformation is expedited by the L\"owdin procedure carried out to the desired order in $\ensuremath{\rho}\ensuremath{\omega}$. However, a Schmidt orthogonalization scheme might prove more advantageous in specific cases. Field-theoretic techniques (of both perturbative and nonperturbative nature) may be brought to bear in the diagonalization of (
Hmn). The inverse of the unitary transformation which takes the basis ${{\ensuremath{\Phi}}_{\mathrm{m}}}$ into the basis ${{\ensuremath{\Theta}}_{\mathrm{m}}}$, applied to the given singular Hamiltonian, leads---for any
F in the class considered---to a non-singular "effective Hamiltonian." Following this formal cue, we set out to construct explicitly a second-quantized effective Hamiltonian which generates the same matrix elements, to prescribed order in $\ensuremath{\rho}\ensuremath{\omega}$, when operating between independent-particle kets of the model occupation-number representation, as does the given Hamiltonian in the configuration-space representation, between the corresponding correlated functions of the L\"owdin basis. The quasiparticles described by this effective Hamiltonian interact via well-behaved two-body, three-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{},
Q-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{} potentials, in contrast to the highly singular two-body interactions of the real particles. For small $|\ensuremath{\rho}\ensuremath{\omega}|$, the effective two-body forces predominate. An avenue to deeper understanding of the applicability of the nuclear-shell model and the dressed-quasiparticle theory of liquid
He3 has been opened. This article treats only the formalism for a uniform extended medium. Numerical calculations, as well as the extension of the method to finite systems, will follow.