J. A. Berger

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

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 non-linear 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 Greens function does not unambiguously determine the underlying system, given in terms of its potential, or non-interacting Greens 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 strong-correlation 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.

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 Laboratoire des Solides Irradiés, Ecole Polytechnique, CNRS, CEA/DSM and European Theoretical Spectroscopy Facility (ETSF), 91128 Palaiseau, France. Laboratoire de Physique Théorique, CNRS, IRSAMC, Université Toulouse III Paul Sabatier and European Theoretical Spectroscopy Facility (ETSF), 118 Route de Narbonne, F-31062 Toulouse Cedex, France Laboratoire 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 (Dated: March 17, 2017)

Solution to the many-body problem in one point

In this work we determine the one-body Greenʼs function as solution of a set of functional integro-differential equations, which relate the one-particle Greenʼs function to its functional derivative with respect to an external potential. In the same spirit as Lani et al (2012 New J. Phys. 14 013056), we do this in a one-point model, where the equations become ordinary differential equations (DEs) and, hence, solvable with standard techniques. This allows us to analyze several aspects of these DEs as well as of standard methods for determining the one-body Greenʼs function that are important for real systems. In particular: (i) we present a strategy to determine the physical solution among the many mathematical solutions; (ii) we assess the accuracy of an approximate DE related to the +cumulant method by comparing it to the exact physical solution and to standard approximations such as ; (iii) we show that the solution of the approximate DE can be improved by combining it with a screened interaction in the random-phase approximation. (iv) We demonstrate that by iterating the Dyson equation one does not always converge to a solution and we discuss which iterative scheme is the most suitable to avoid such errors.

Green Functions and Self-Consistency: Insights From the Spherium Model

We report an exhaustive study of the performance of different variants of Green function methods for the spherium model in which two electrons are confined to the surface of a sphere and interact via a genuine long-range Coulomb operator. We show that the spherium model provides a unique paradigm to study electronic correlation effects from the weakly correlated regime to the strongly correlated regime, since the mathematics are simple while the physics is rich. We compare perturbative GW, partially self-consistent GW and second-order Green function (GF2) methods for the computation of ionization potentials, electron affinities, energy gaps, correlation energies as well as singlet and triplet neutral excitations by solving the Bethe-Salpeter equation (BSE). We discuss the problem of self-screening in GW and show that it can be partially solved with a second-order screened exchange correction (SOSEX). We find that, in general, self-consistency deteriorates the results with respect to those obtained within perturbative approaches with a Hartree-Fock starting point. Finally, we unveil an important problem of partial self-consistency in GW: in the weakly correlated regime, it can produce artificial discontinuities in the self-energy caused by satellite resonances with large weights.