Self-Consistent Green’s Function Approaches

Walter Tarantino, Pina Romaniello, J. A. Berger +1 more

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.

H. Y. Chu, Z. Ye, Hiroomi Umezawa

Abstract We study the properties of a dissipative two‐state fermionic system embedded in a phonon bath at zero temperature. In contrast to the traditional path‐integral and canonical transformation approaches, we propose a self‐consistent Green's function approach. Our results show that the mean field theory overestimates the polarization of the system. To justify our approach, we consider a simplest case in which there is only one phonon mode. This simple model can be ‘exactly’ diagonalized. We find that the results from the self‐consistent Green's function approach agree well with the quantum mechanical diagonalization results.