A Compact Treatment of Singular Impurities Using the Artificial Friedel Resonance (FAIR) Technique

Abstract

Building on the tools that Friedel introduced in the 1950s, an off-spring of the Friedel resonance is developed, which is called the FAIR approach for “Friedel Artificially Inserted Resonance”. In the FAIR approach, an arbitrary s-electron state $a_{0}^{\dagger}$ is cut out of the conduction band, and the remaining free electron Hamiltonian is orthogonalized, yielding an artificial Friedel resonance. In the presence of a real Friedel d-resonance state d †, one can find an optimal fair state $a_{0}^{\dagger}$ so that the exact n-electron ground state consists of two Slater states, one containing the d-electron and the other the fair state $a_{0}^{\dagger}$ . This separation according to the d-occupation is ideal for impurities with a Coulomb interaction between d-electrons. The wave function of the Friedel–Anderson (FA) impurity in the magnetic state and the singlet state are constructed with the FAIR method using two fair states $a_{0}^{\dagger}$ and $b_{0}^{\dagger}$ . The magnetic state consists of four, and the singlet state consists of eight Slater states. The latter is invariant with respect to the inversion of all spins. The FAIR ground state Ψ SS for the singlet state is composed of two magnetic states Ψ MS with opposite moments which are not orthogonal to each other. The degree of overlap determines the Kondo energy. Because of the compactness of the wave function, a number of properties can be calculated relatively quickly. Comparison with the best previous ground state energy and d-occupation by Gunnarson and Schoenhammer show excellent agreement. A number of physical properties are calculated; among them are (i) the Kondo cloud in a small magnetic field. Two components of polarization are observed (in linear response), an oscillating part and a nonoscillating part. (ii) The fidelity which represents the scalar product between the ground states of the symmetric FA-impurity and the symmetric Friedel impurity is calculated as a function of the number of electron states. (In the literature, this has the somewhat misleading name of fidelity.) This calculation does not show an Anderson orthogonality catastrophe, which indicates that the FA ground state has a phase shift of π/2 at the Fermi energy. (iii) The Friedel oscillations of the FA-impurity. Surprisingly, these oscillations are very similar to the Friedel oscillations of a very narrow Friedel resonance at the Fermi level. The amplitude A(r) is close to zero at short distances and saturates at two for large distances. The inversion point where A(r)=1 correlates with the characteristic energies of the impurities, the half-band width Γ for the Friedel impurity and the Kondo energy for the FA-impurity. This raises the fascinating question how these simple properties are hidden in the multielectron wave function.