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Bulletin of the American Physical Society

The exact solution of the many-body problem in one-point: insights in approximate Green's function approaches

J. A. Berger, Pina Romaniello, Lucia Reining

Published Mar 4, 2014• 0 citations
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Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on many-body perturbation theory. They can be obtained by iterating a set of functional differential equations (DEs) relating the one-particle Green's function (GF) to its functional derivative with respect to an external perturbing potential. In this work, we apply a linear response expansion in order to obtain insights into various approximations for GF calculations. The expansion leads to an effective screening while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations, we discretize them and retain only one point in space, spin and time for all variables. Within this one-point model we obtain an explicit solution for the GF, which allows us to explore the structure of the general family of solutions and to determine the specific solution that corresponds to the physical one. Moreover, we analyze the performances of established approaches like GW over the whole range of interaction strength, and we explore alternative approximations. Finally, we link certain approximations for the exact solution to the corresponding manipulations of the DE which produces them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the DE is known.
The many-body problem cannot in general be solved exactly, and one of the most prominent approximations is to build perturbation expansions. A huge variety of expansions are possible, which differ by the quantity to be expanded, the expansion variable, the starting point, and ideas of how to re-sum or terminate the series. Although much has been discussed and much has been done, some choices were made for historical reasons, in particular, limited computation or storage capacities. The present work aims to examine the justification for different choices made in different contexts, by comparing the components of functionals based on Green's functions on the one side, and on the charge density on the other side. Of particular interest will be the question of how to build an optimal starting point for the approximation of non-local quantities, making use of near- or far-sightedness, and daring to consider models beyond the homogeneous electron gas. This will include the use of connector approximations. We will also discuss why it is a good idea to build functionals of the density.
Abstract The energy gap of correlated Hubbard clusters is well studied for one‐dimensional systems using analytical methods and density‐matrix‐renormalization‐group (DMRG) simulations. Beyond 1D, however, exact results are available only for small systems by quantum Monte Carlo. For this reason and, due to the problems of DMRG in simulating 2D and 3D systems, alternative methods such as Green functions combined with many‐body approximations (GFMBA), that do not have this restriction, are highly important. However, it has remained open whether the approximate character of GFMBA simulations prevents the computation of the Hubbard gap. Here we present new GFMBA results that demonstrate that GFMBA simulations are capable of producing reliable data for the gap which agrees well with the DMRG benchmarks in 1D. An interesting observation is that the accuracy of the gap can be significantly increased when the simulations give up certain symmetry restriction of the exact system, such as spin symmetry and spatial homogeneity. This is seen as manifestation and generalization of the “symmetry dilemma” introduced by Löwdin for Hartree–Fock wave function calculations.