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Physical review. B, Condensed matter

Anderson impurity model in the cumulant approach

G. Polatsek, K. W. Becker, Vilen Zevin

Published Sep 15, 1991• 1 citation

Abstract

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 ${\mathit{N}}_{\mathit{f}}$\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 ${\mathit{N}}_{\mathit{f}}$ and U both finite our procedure leads to the result of the variational approach of Gunnarsson and Sch\"onhammer.

DOI: 10.1103/physrevb.44.6062

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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

{\mathit{N}}_{\mathit{f}}

\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

{\mathit{N}}_{\mathit{f}}

and U both finite our procedure leads to the result of the variational approach of Gunnarsson and Sch"onhammer.

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.

An approximate ground state of the Anderson-Friedel impurity problem is presented in a very compact form. It requires solely the optimization of two localized electron states and consists of four Slater states (Slater determinants). The resulting singlet ground-state energy lies far below the Anderson mean-field solution and agrees well with the numerical results by Gunnarsson and Sch\"onhammer, who used an extensive

1∕{N}_{f}

expansion for a spin-

\frac{1}{2}

impurity with double occupancy of the impurity level.

Building an effective Hamiltonian utilizing a projector-operatorprocedure, we derive an approximation based on a selfconsistenthybridization expansion to study the ground stateproperties of the Anderson Impurity model.We applied the approximationto the general case of finite Coulomb repulsionU, extending previous work with the same formalism in theinfinite-U case. The ground state energy and their related zerotemperature properties are accurately obtained in the case inwhich U is large enough, but still finite, as compared with the rest of energy scales involved in the model. The results for thevalence of the impurity are compared with exact results thatwe obtain from equations derived using the Bethe ansatz andwith a perturbative approach. The magnetization and magneticsusceptibility is also compared with Bethe ansatz results. Inorder to do this comparison, we also show how to regularizethe Bethe ansatz integral equations necessary to calculate theimpurity valence, for arbitrary values of the parameters.

We consider the cumulant expansion of the PAM employing the hybridization as perturbation (Phys. Rev. B 50, 17933 (1994)), and we obtain formally exact one-electron Green's functions (GF). These GF contain effective cumulants that are as difficult to calculate as the original GF, and the Atomic Approximation consists in substituting the effective cumulants by the ones that correspond to the atomic case, namely by taking a conduction band of zeroth width and local hybridization. This approximation has already been used for the case of infinite electronic repulsion U (Phys. Rev. B 62, 7882 (2000)), and here we extend the treatment to the case of finite U. The method can also be applied to the single impurity Anderson model (SIAM), and we give explicit expressions of the approximate GF both for the PAM and the SIAM.

A rational representation for the self-energy is explored to interpolate the solution of the Anderson impurity model in the general orbitally degenerate case. Several constraints such as Friedel's sum rule and the positions of the Hubbard bands, as well as the value of the quasiparticle residue, are used to establish the equations for the coefficients of the interpolation. We employ two fast techniques, the slave-boson mean-field and the Hubbard I approximations, to determine the functional dependence of the coefficients on doping, degeneracy, and the strength of the interaction. The obtained spectral functions and self-energies are in good agreement with the results of the numerically exact quantum Monte Carlo method.

The twofold degenerate Anderson impurity model [1-4] is investigated in normal and superconducting states, and the strong electronic correlations of d-electrons of impurity ion are taken into account by elaborating a suitable diagram technique. We discuss the properties of the Slater-Kanamori model [2-4] of d-impurity electrons. After finding the eigenfunctions and eigenvalues of all 16 local states, we determine the local one-particle propagator. Then we construct the perturbation theory around the atomic limit of the impurity ion and obtain the Dyson type equations between impurity electron propagators and normal and anomalous correlation functions. By summing infinite series of ladder diagrams, the approximation for correlation functions is established. The criterion for appearance of a superconducting state of the model is discussed.