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New Journal of Physics

Solution to the many-body problem in one point

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

Published Nov 11, 2014• 25 citations
Publisher

Abstract

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.

DOI
10.1088/1367-2630/16/11/113025

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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.
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.
Many-body perturbation theory is often formulated in terms of an expansion in the dressed instead of the bare Green's function, and in the screened instead of the bare Coulomb interaction. However, screening can be calculated on different levels of approximation, and it is important to define what is the most appropriate choice. We explore this question by studying a zero-dimensional model (so called 'one-point model') that retains the structure of the full equations. We study both linear and non-linear response approximations to the screening. We find that an expansion in terms of the screening in the random phase approximation is the most promising way for an application in real systems. Moreover, by making use of the nonperturbative features of the Kadanoff-Baym equation for the one-body Green's function, we obtain an approximate solution in our model that is very promising, although its applicability to real systems has still to be explored.
In a recent letter [Riva et al., Phys. Rev. Lett. 131, 216401 (2023)] we presented the multichannel Dyson equation (MCDE), in which two or more many-body Green's functions are coupled. In this work we will give further details of the MCDE approach. In particular we will discuss (1) the derivation of the MCDE and the definition of the space in which it is to be solved, (2) the rationale of the approximation to the multichannel self-energy, (3) a diagrammatic analysis of the MCDE, and (4) the recasting of the MCDE on an eigenvalue problem with an effective Hamiltonian that can be solved using standard numerical techniques. This work mainly focuses on the coupling between the one-body Green's function and the three-body Green's function to describe photoemission spectra, but the MCDE method can be generalized to the coupling of other many-body Green's functions and to other spectroscopies.
We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.
In a recent letter [Phys. Rev. Lett. 131, 216401] we presented the multichannel Dyson equation (MCDE) in which two or more many-body Green's functions are coupled. In this work we will give further details of the MCDE approach. In particular we will discuss: 1) the derivation of the MCDE and the definition of the space in which it is to be solved; 2) the rationale of the approximation to the multichannel self-energy; 3) a diagrammatic analysis of the MCDE; 4) the recasting of the MCDE on an eigenvalue problem with an effective Hamiltonian that can be solved using standard numerical techniques. This work mainly focuses on the coupling between the one-body Green's function and the three-body Green's function to describe photoemission spectra, but the MCDE method can be generalized to the coupling of other many-body Green's functions and to other spectroscopies.
We present results for many-body perturbation theory for the one-body Green's function at finite temperatures using the Matsubara formalism. Our method relies on the accurate representation of the single-particle states in standard Gaussian basis sets, allowing to efficiently compute, among other observables, quasiparticle energies and Dyson orbitals of atoms and molecules. In particular, we challenge the second-order treatment of the Coulomb interaction by benchmarking its accuracy for a well-established test set of small molecules, which includes also systems where the usual Hartree-Fock treatment encounters difficulties. We discuss different schemes how to extract quasiparticle properties and assess their range of applicability. With an accurate solution and compact representation, our method is an ideal starting point to study electron dynamics in time-resolved experiments by the propagation of the Kadanoff-Baym equations.