Solution to the many-body problem in one point

Berger, J. A.; Romaniello, Pina; Tandetzky, Falk; Mendoza, Bernardo S.; Brouder, Christian; Reining, Lucia

doi:10.1088/1367-2630/16/11/113025

Cited by 24 publications

(40 citation statements)

References 19 publications

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“…In the present work we use the OPM without approximations, which simulates the full many-body problem. The exact OPM Green's function was derived in [15] from the one-point equivalent of the equation of motion of G, expressed as a functional differential equation [16]. In equation (1)s is given as a functional of the bare interaction u and the noninteracting Green's function y 0 .…”

Section: Theory and Discussionmentioning

confidence: 99%

“…This sign problem is a priori a disaster because there is no unique prescription of how to avoid unphysical solutions. The OPM highlights the reducible polarizability [15]…”

Section: Tddft and The Map C C mentioning

confidence: 99%

“…To lowest order, this corresponds to HF, =s uy, HF for strong interaction. We call this functional strong-interaction HF (SIN-HF) 15 . Both self-energies yield two solutions.…”

Section: Tddft and The Map C C mentioning

confidence: 99%

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Unphysical and physical solutions in many-body theories: from weak to strong correlation

et al. 2015

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Many-body theory is largely based on self-consistent equations that are constructed in terms of the physical quantity of interest itself, for example the density. Therefore, the calculation of important properties such as total energies or photoemission spectra requires the solution of nonlinear equations that have unphysical and physical solutions. In this work we show in which circumstances one runs into an unphysical solution, and we indicate how one can overcome this problem. Moreover, we solve the puzzle of when and why the interacting Green's function does not unambiguously determine the underlying system, given in terms of its potential, or non-interacting Green's function. Our results are general since they originate from the fundamental structure of the equations. The absorption spectrum of lithium fluoride is shown as one illustration, and observations in the literature for some widely used models are explained by our approach. Our findings apply to both the weak and strongcorrelation regimes. For the strong-correlation regime we show that one cannot use the expressions that are obtained from standard perturbation theory, and we suggest a different approach that is exact in the limit of strong interaction.

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Section: Theory and Discussionmentioning

confidence: 99%

“…This sign problem is a priori a disaster because there is no unique prescription of how to avoid unphysical solutions. The OPM highlights the reducible polarizability [15]…”

Section: Tddft and The Map C C mentioning

confidence: 99%

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Unphysical and physical solutions in many-body theories: from weak to strong correlation

et al. 2015

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“…), multi‐dimensional, functional integro‐differential equation. Even worse, such an equation can have many solutions, and it is not obvious how to select the physical one (for a discussion, see, e.g, Ref ).…”

Section: The Gw Approximation: Theorymentioning

confidence: 99%

The GW approximation: content, successes and limitations

Reining

2017

WIREs Comput Mol Sci

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Many observables such as the density, total energy, or electric current, can be expressed explicitly in terms of the one‐body Green's function, which describes electron addition or removal to or from a system. An efficient way to determine such a Green's function is to introduce a self‐energy, which is a nonlocal and dynamic effective potential that influences the propagation of particles in an interacting system. The state‐of‐the art approximation for the self‐energy is the GW approximation, where the system to (or from) which the electron is added (or removed) is described as a polarizable, screening, medium. This is expressed by the name of the approximation: ‘GW’ stands for the one‐body Green's function G and for W, the dynamically screened Coulomb interaction. The GW approximation is very popular for the calculation of band structures in solids, and increasingly used also to describe nanostructures, clusters, and molecules. As compared to static mean‐field approximations for the effective potential, the dynamical screening of the Coulomb interaction in GW leads to a renormalization of energies, to broadening and/or to the observation of additional excitations. An analysis of the approximations that lead to the GW self‐energy, and of the underlying picture, explains the successes and the limitations of the approach. This article is categorized under: Electronic Structure Theory > Density Functional Theory Electronic Structure Theory > Ab Initio Electronic Structure Methods Theoretical and Physical Chemistry > Spectroscopy Structure and Mechanism > Computational Materials Science

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“…Schäfer et al [7,9] and first introduced by Stan and collaborators [8] in the context of the so called One Point Model (OPM) [22,23], which is a mathematically simplified framework for studying the functional formulation of the Green's function formalism. (see also [24,25] (11) and (14).…”

Section: Explicit Functionalsmentioning

confidence: 99%

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

et al. 2017

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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 non-interacting 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 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]. 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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Solution to the many-body problem in one point

Cited by 24 publications

References 19 publications

Unphysical and physical solutions in many-body theories: from weak to strong correlation

Unphysical and physical solutions in many-body theories: from weak to strong correlation

The GW approximation: content, successes and limitations

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

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