Self-consistent Dyson equation and self-energy functionals: An analysis and illustration on the example of the Hubbard atom

Walter Tarantino, Pina Romaniello, J. A. Berger, and Lucia Reining
Phys. Rev. B 96, 045124 – Published 18 July 2017

Abstract

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.

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  • Received 17 March 2017
  • Revised 27 June 2017

DOI:https://doi.org/10.1103/PhysRevB.96.045124

©2017 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Walter Tarantino1,*, Pina Romaniello2, J. A. Berger3, and Lucia Reining1

  • 1Laboratoire des Solides Irradiés, Ecole Polytechnique, CNRS, CEA, Université Paris-Saclay, and European Theoretical Spectroscopy Facility (ETSF), F-91128 Palaiseau, France
  • 2Laboratoire de Physique Théorique, IRSAMC, CNRS, Université Toulouse III - Paul Sabatier and European Theoretical Spectroscopy Facility (ETSF), 118 Route de Narbonne, F-31062 Toulouse Cedex, France
  • 3Laboratoire de Chimie et Physique Quantiques, IRSAMC, Université Toulouse III - Paul Sabatier, CNRS and European Theoretical Spectroscopy Facility (ETSF), 118 Route de Narbonne, F-31062 Toulouse Cedex, France

  • *walter.tarantino@polytechnique.edu

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Vol. 96, Iss. 4 — 15 July 2017

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