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arXiv:cond-mat/0602196 · Physical Review B

Compact approximate solution to the Friedel-Anderson impurity problem

Gerd Bergmann

Published Oct 23, 2006• 15 citations
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Abstract

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.

DOI
10.1103/physrevb.74.144420

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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∕Nf1∕{N}_{f}1∕Nf​ expansion for a spin-12\frac{1}{2}21​ impurity with double occupancy of the impurity level.
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.
By means of a projector operator formalism we study the ground state properties of the Anderson Impurity Hamiltonian. The non-perturbative treatment of the model agrees with the previous one, obtained by Inagaki [Prog. Theor. Phys. 62, 1441 (1979)] by means of a perturbation expansion with respect the hybridization term. We go beyond the Inagaki's formalism to the next leading order. It provides a very accurate calculation of the energy spectrum in the total spin sector and, in particular, the ground state energy in the whole parameter space. For a one body spinless system, the dependence of the ground state energy as a function of the impurity level obtained by this procedure remarkably agrees with analytical results. For the many body case the occupancy of the impurity as a function of the parameters is studied and it agrees with the corresponding one obtained by using the Bethe ansatz and the Numerical Renormalization Group solution of the Hamiltonian. The magnetization and susceptibility of the impurity is analyzed by studying the response of the system to an external magnetic field, from which it is possible to extract the Kondo temperature. The dependence of the Kondo scale with the parameters of the model is in excellent agreement with well-known results.
The variational approach of Gunnarsson and Sch\"onhammer to the Anderson impurity model is generalized to study d-wave superconductors in the presence of dilute spin-12\frac{1}{2}21​ impurities. We show that the local moment is screened when the hybridization exceeds a nonzero critical value at which the ground state changes from a spin doublet to a spin singlet. The electron spectral functions are calculated in both phases. We find that while a Kondo resonance develops above the Fermi level in the singlet phase, the spectral function exhibits a low-energy spectral peak below the Fermi level in the spin-doublet phase. The origin of such a ``virtual Kondo resonance'' is the existence of low-lying collective excitations in the spin-singlet sector. We discuss our results in connection with recent spectroscopic experiments in Zn-doped high-Tc{T}_{c}Tc​ superconductors.
The Anderson impurity model for finite values of the Coulomb repulsion U is studied using a slave-boson representation for the empty and doubly occupied f level. In order to avoid well-known problems with a naive mean-field theory for the boson fields, we use the coherent-state-path integral representation to first integrate out the double-occupancy slave bosons. The resulting effective action is linearized using two-time auxiliary fields. After integration over the fermionic degrees of freedom, one obtains an effective action suitable for a 1/Nf{\mathit{N}}_{\mathit{f}}Nf​ expansion. The same problem remains concerning the constraint as in the infinite U case. For T\ensuremath{\rightarrow}0 and Nf{\mathit{N}}_{\mathit{f}}Nf​\ensuremath{\rightarrow}\ensuremath{\infty}, exact results for the ground-state properties are recovered in the saddle-point approximation. Numerical solutions of the saddle-point equations show that even in the spin-degenerate case Nf{\mathit{N}}_{\mathit{f}}Nf​=2 the results are quite good.