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Physical review. B, Condensed matter

Equation-of-motion treatment of the impurity Anderson model with a finite on-site Coulomb repulsion

Kicheon Kang, Byung Il Min

Published Oct 15, 1995• 28 citations
Publisher

Abstract

The impurity Anderson model with a finite Coulomb repulsion U has been studied by using the conventional equation-of-motion treatment for the f-electron Green's function. To get a closed equation, two types of approximations are incorporated. One is to neglect, in a large U limit, higher order Green's functions which involve three f-electron annihilation operators. The other is to linearize the equation of motion by using a projection operator method. The average valency ${\mathit{n}}_{\mathit{f}}$ and the spectral density of states (DOS) of f electrons are evaluated from the self-consistent equation for the Green's function. Results for ${\mathit{n}}_{\mathit{f}}$ are in good agreement with exact results of Bethe ansatz method, and the obtained spectral DOS exhibits a well-known three-peak structure. Effects of finite U and finite temperature on the spectral DOS are discussed.

DOI
10.1103/physrevb.52.10689

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The impurity Anderson model with a finite Coulomb repulsion U has been studied by using the conventional equation-of-motion treatment for the f-electron Green's function. To get a closed equation, two types of approximations are incorporated. One is to neglect, in a large U limit, higher order Green's functions which involve three f-electron annihilation operators. The other is to linearize the equation of motion by using a projection operator method. The average valency nf{\mathit{n}}_{\mathit{f}}nf​ and the spectral density of states (DOS) of f electrons are evaluated from the self-consistent equation for the Green's function. Results for nf{\mathit{n}}_{\mathit{f}}nf​ are in good agreement with exact results of Bethe ansatz method, and the obtained spectral DOS exhibits a well-known three-peak structure. Effects of finite U and finite temperature on the spectral DOS are discussed.
We discuss the ground-state energy of the Anderson degenerate impurity model for both finite and infinite on-site Coulomb repulsion U. The goal of the paper is to demonstrate the applicability of a theoretical approach to static ground-state properties introduced recently. This approach is based on the projection technique and on the introduction of cumulants. It allows for proper treatment of strong electronic correlations that are incorporated in the model. It also guarantees the size consistency of the evaluated physical quantities. We start from a compact expression for the ground-state energy, which is expanded into a continued fraction. By summing up the infrared divergencies, a self-consistent equation for the ground-state energy is found. For degeneracy Nf{\mathit{N}}_{\mathit{f}}Nf​\ensuremath{\rightarrow}\ensuremath{\infty} and U\ensuremath{\rightarrow}\ensuremath{\infty} the equation can be solved rigorously and gives the well-known Varma-Yafet solution. For Nf{\mathit{N}}_{\mathit{f}}Nf​ and U both finite our procedure leads to the result of the variational approach of Gunnarsson and Sch\"onhammer.
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.
The paper addresses the many-body quasiparticle dynamics of the Anderson impurity model at finite temperatures in the framework of the equation-of-motion method. We find a new exact identity relating the one-particle and many-particle Green’s functions. Using this identity we present a consistent and general scheme for a construction of generalized mean fields (elastic scattering corrections) and self-energy (inelastic scattering) in terms of Dyson equation. A new approach for the complex expansion for the single-particle propagator in terms of Coulomb repulsion U and hybridization V is proposed. Using the exact identity, the essentially new many-body dynamical solution of SIAM has been derived. This approach offers a new way for the systematic construction of the approximative interpolating dynamical solutions of the strongly correlated electron systems.
We present self-consistent Green-function calculations for the properties of the symmetric Anderson impurity model at zero temperature. A stable method for calculating the spectral density directly on the real frequency axis in the spin-fluctuation limit is described. Results using an approximation to the self-energy, previously termed the fluctuation-exchange approximation, which includes the effects of spin and density fluctuations (within the random-phase approximation) and multiple electron-electron scattering are compared with second-order perturbation-theory calculations. The static spin and charge susceptibilities, calculated in a manner consistent with conservation laws, are compared with the exact results from the Bethe ansatz. We find that the fluctuation-exchange approximation gives a good description of the charge susceptibility over a wide range of interaction strengths and is a great improvement over the self-consistent second-order theory. However, the T-linear coefficient of the specific heat is greatly overestimated by the fluctuation-exchange approximation and this leads to the underestimation of the spin susceptibility. The self-consistent second-order perturbation theory, which underestimates the specific-heat coefficient, predicts a magnetic ground state for sufficiently high interaction strengths.
A review of electronic dynamics of single-impurity and many-impurity Anderson models is contained in this report. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the non-perturbative technique for the thermodynamic Green functions. The method of the irreducible Green functions is used as the basic tool. This irreducible Green functions method allows one to describe the quasiparticle spectra with damping of the strongly correlated electron systems in a very general and natural way and to construct the relevant dynamical solution in a self-consistent way on the level of Dyson equation without decoupling the chain of the equations of motion for the Green functions. The subject matter includes the improved interpolating solution of the Anderson model. It was shown that an interpolating approximation, which simultaneously reproduces the weak-coupling limit up to second order in the interaction strength U and the strong-coupling limit up to second order in the hybridization V (and thus also fulfils the atomic limit) can be formulated self-consistently. This approach offers a new way for the systematic construction of approximate interpolation dynamical solutions of strongly correlated electron systems.
A new approach to many-body quasiparticle dynamics of the Anderson impurity model at flnite temperatures is proposed in the framework of the equation-of-motion method.We flnd a new exact identity, which relates the one-particle and many-particle Green functions.Using this identity, it is shown that a new interpolating approximation for the one-particle Green function can be constructed.The approximation which simultaneously reproduces tle weak-coupling limit and the strong coupling limit can be formulated self-consistently.This approach offers a new method for a systematic construction of the approximate interpolating dynamical solutions of strongly correlated electron models.