Perturbation theory using self-consistent Green's functions is one of the most widely used approaches to study many-body effects in condensed matter. On the basis of general considerations and by performing analytical calculations for the specific example of the Hubbard atom, we discuss some key features of this approach. We show that when the domain of the functionals that are used to realize the map between the noninteracting and the interacting Green's functions is properly defined, there exists a class of self-energy functionals for which the self-consistent Dyson equation has only one solution, which is the physical one. We also show that manipulation of the perturbative expansion of the interacting Green's function may lead to a wrong self-energy as a functional of the interacting Green's function, at least for some regions of the parameter space. These findings confirm and explain numerical results of Kozik et al. for the widely used skeleton series of Luttinger and Ward [Phys. Rev. Lett. 114, 156402 (2015)]. Our study shows that it is important to distinguish between the maps between sets of functions and the functionals that realize those maps. We demonstrate that the self-consistent Green's functions approach itself is not problematic, whereas the functionals that are widely used may have a limited range of validity.Perturbation theory using self-consistent Green's functions is one of the most widely used approaches to study many-body effects in condensed matter. On the basis of general considerations and by performing analytical calculations for the specific example of the Hubbard atom, we discuss some key features of this approach. We show that when the domain of the functionals that are used to realize the map between the noninteracting and the interacting Green's functions is properly defined, there exists a class of self-energy functionals for which the self-consistent Dyson equation has only one solution, which is the physical one. We also show that manipulation of the perturbative expansion of the interacting Green's function may lead to a wrong self-energy as a functional of the interacting Green's function, at least for some regions of the parameter space. These findings confirm and explain numerical results of Kozik et al. for the widely used skeleton series of Luttinger and Ward [Phys. Rev. Lett. 114, 156402 (2015)]. Our study shows that it is important to distinguish between the maps between sets of functions and the functionals that realize those maps. We demonstrate that the self-consistent Green's functions approach itself is not problematic, whereas the functionals that are widely used may have a limited range of validity.

arXiv:1703.05587Physical review. B./Physical review. BJul 18, 201755 citations

A Scalable Numerical Approach to the Solution of the Dyson Equation for the Non‐Equilibrium Single‐Particle Green's Function

Natale Walter Talarico, Sabrina Maniscalco, Nicola Lo Gullo

A numerical method to solve the set of Dyson‐like equations in the framework of non‐equilibrium Green's functions is presented. The technique is based on the self‐consistent solution of the Dyson equations for the interacting single‐particle Green's function once a choice for the self‐energy, functional of the single‐particle Green's function itself, is made. The authors briefly review the theory of the non‐equilibrium Green's functions in order to highlight the main point useful in discussing the proposed technique. Then, the numerical implementationis presented and discussed, which is based on the distribution of the Keldysh components of the Green's function and the self‐energy on a grid of processes. It is discussed how the structure of the considered self‐energy approximations influences the distribution of the matrices in order to minimize the communication time among processes and which should be considered in the case of other approximations. The authors give an example of the application of this technique to the case of quenches in ultracold gases and to the single impurity Anderson model, also discussing the convergence and the stability features of the approach.A numerical method to solve the set of Dyson‐like equations in the framework of non‐equilibrium Green's functions is presented. The technique is based on the self‐consistent solution of the Dyson equations for the interacting single‐particle Green's function once a choice for the self‐energy, functional of the single‐particle Green's function itself, is made. The authors briefly review the theory of the non‐equilibrium Green's functions in order to highlight the main point useful in discussing the proposed technique. Then, the numerical implementationis presented and discussed, which is based on the distribution of the Keldysh components of the Green's function and the self‐energy on a grid of processes. It is discussed how the structure of the considered self‐energy approximations influences the distribution of the matrices in order to minimize the communication time among processes and which should be considered in the case of other approximations. The authors give an example of the application of this technique to the case of quenches in ultracold gases and to the single impurity Anderson model, also discussing the convergence and the stability features of the approach.

arXiv:1809.10111physica status solidi (b)Mar 8, 201916 citations

Fully self-consistent solution of the Dyson equation using a plane-wave basis set

Lin‐Wang Wang

Self-consistent solutions of the Dyson equation are obtained using a plane-wave basis set for seven small molecules. Such self-consistent solutions can help to unify the different $GW$ self-consistent schemes, reduce the scatter of results in current $GW$ calculations, and shed light on the true effects of $GW$ self-consistency. Unlike other works of self-consistent $GW$ calculations, in the present work the Green's function is expressed as a matrix under the plane-wave basis set. The algorithmic details which enable such calculations are presented. The ability to solve the full Green's function using a plane-wave basis set may open the door for future beyond- $GW$ many-body perturbation theory calculations.Self-consistent solutions of the Dyson equation are obtained using a plane-wave basis set for seven small molecules. Such self-consistent solutions can help to unify the different $GW$ self-consistent schemes, reduce the scatter of results in current $GW$ calculations, and shed light on the true effects of $GW$ self-consistency. Unlike other works of self-consistent $GW$ calculations, in the present work the Green's function is expressed as a matrix under the plane-wave basis set. The algorithmic details which enable such calculations are presented. The ability to solve the full Green's function using a plane-wave basis set may open the door for future beyond-$GW$ many-body perturbation theory calculations.

Physical Review BMar 23, 201514 citations

Self-consistent solutions of the Dyson equation are obtained using a plane-wave basis set for seven small molecules. Such self-consistent solutions can help to unify the different

GW

self-consistent schemes, reduce the scatter of results in current

GW

calculations, and shed light on the true effects of

GW

self-consistency. Unlike other works of self-consistent

GW

calculations, in the present work the Green's function is expressed as a matrix under the plane-wave basis set. The algorithmic details which enable such calculations are presented. The ability to solve the full Green's function using a plane-wave basis set may open the door for future beyond-

GW

many-body perturbation theory calculations.