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Progress of Theoretical Physics

A Solution of a Dyson Equation in Many-Impurity Problems in Solids

Shôzô Takeno

Published Nov 1, 1968• 34 citations
Publisher

Abstract

An approximate solution of a Dyson equation satisfied by a single particle Green's function in many-impurity problems in solids is obtained for an extended impurity perturbation potential. We assume that there is no overlapping of any two impurity potentials and that the range of the impurity potential is arbitrary. The coordinate representation of the Dyson equation is solved by transforming an iterative solution of the equation into a form with no repetitive summation indices appearing in any term of given order. By neglecting fluctuations of the impurity distribution, a set of equations are derived to determine the Green's function and the self-energy in a self-consistent manner. A group-theoretic consideration of the self-energy and some sum rules are given.

DOI
10.1143/ptp.40.942

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An approximate solution of a Dyson equation satisfied by a single particle Green's function in many-impurity problems in solids is obtained for an extended impurity perturbation potential. We assume that there is no overlapping of any two impurity potentials and that the range of the impurity potential is arbitrary. The coordinate representation of the Dyson equation is solved by transforming an iterative solution of the equation into a form with no repetitive summation indices appearing in any term of given order. By neglecting fluctuations of the impurity distribution, a set of equations are derived to determine the Green's function and the self-energy in a self-consistent manner. A group-theoretic consideration of the self-energy and some sum rules are given.
A new many-body theory perturbationally consistent to all orders of the perturbation is presented and applied to the Hubbard problem as an example. Use is made of the recent Kim and Wilson commutator projection operator to derive the Dyson equation for the one-particle Green's function and an exact set of coupled differential equations for the damping or self-energy term which are solved self-consistently. The Hubbard III result follows in our first approximation and the Esterling-Lange (EL)-type result in the second. Since we keep the hopping finite throughout the calculation, no degeneracy problem is encountered as in EL.
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.
A new approach to the one-particle Green's functions G\mathit{G}G for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part M\mathit{M}M related to G\mathit{G}G via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematical way a set of approximation schemes (nthn\mathrm{th}nth-order ADC schemes) that represent infinite partial summations for M\mathit{M}M and (via the Dyson equation) for G\mathit{G}G being complete through nthn\mathrm{th}nth order of perturbation theory. The corresponding mathematical procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed. While the second- and third-order schemes can be viewed as systematic rederivations of previous approximation schemes, the fourth-order ADC scheme represents a complete fourth-order approximation for the self-energy and the one-particle Green's function which was hitherto not available.
A numerical method to solve the set of Dyson‐like equations in the framework of non‐equilibrium Green's functions is presented. The technique is based on the self‐consistent solution of the Dyson equations for the interacting single‐particle Green's function once a choice for the self‐energy, functional of the single‐particle Green's function itself, is made. The authors briefly review the theory of the non‐equilibrium Green's functions in order to highlight the main point useful in discussing the proposed technique. Then, the numerical implementationis presented and discussed, which is based on the distribution of the Keldysh components of the Green's function and the self‐energy on a grid of processes. It is discussed how the structure of the considered self‐energy approximations influences the distribution of the matrices in order to minimize the communication time among processes and which should be considered in the case of other approximations. The authors give an example of the application of this technique to the case of quenches in ultracold gases and to the single impurity Anderson model, also discussing the convergence and the stability features of the approach.
Solving the single-impurity Anderson model is a basic problem of solid state physics.The single-impurity Anderson model is very important, at present it is also used for systems with quantum impurities, e.g.semiconductor quantum dots and molecular transistors.Its main application is in the scheme of dynamical mean eld theory describing strong correlation electron systems.To solve the single-impurity Anderson model problem we use the equation of motion Green function approach.In this report we present the novel equation of motion approximation in which we dierentiate the Green function over both time variables.This diers from the commonly used equation of motion solution by Appelbaum, Penn and Lacroix where the authors take time derivative only over primary time variable.After extending calculations to higher order Green functions we nd the new approximate dynamical solution of single-impurity Anderson model.The results are compared with the solutions to the single-impurity Anderson model problem at intermediate Coulomb repulsion U such as the modied iterative perturbation theory.Our approach is suitable for describing quantum dots.