How the Kondo ground state avoids the orthogonality catastrophe

TL;DR

This paper investigates how the Kondo ground state avoids the orthogonality catastrophe by analyzing the multi-electron scalar product in different impurity states, revealing the role of energy separation rather than the number of states.

Contribution

It demonstrates that the Kondo singlet state maintains a finite MESP, preventing the orthogonality catastrophe, and identifies the energy separation as the key factor.

Findings

Magnetic and triplet states show exponential MESP decrease with Wilson states.

Energy separation from the Fermi level determines MESP reduction.

The singlet state maintains a finite MESP, avoiding the catastrophe.

Abstract

In the presence of a magnetic impurity the spin-up and down band states are modified differently by the impurity. If the multi-electron scalar product (MESP) between the occupied spin-up and down states approaches zero then this defines an orthogonality catastrophe. In the present paper the MESP is investigated for the FAIR (Friedel Artificial Iserted Resonance) solution for a Friedel-Anderson impurity. A basis of Wilson states is used. The MESP is numerically determined for the (enforced) magnetic, the singlet, and the triplet states as a function of the number N of Wilson states. The magnetic and the triplet state show an exponentially decreasing MESP as a function of N. Surprisingly it is not the number of states which causes this decrease. It is instead the energy separation of the highest occupied state from the Fermi energy which determines the reduction of the MESP. In the singlet state the ground-state requires a finite MESP to optimize its energy. As a consequence there is no orthogonality catastrophe. The MESP approaches a saturation value as function of N.