Numerical Calculation of the Fidelity for the Kondo and the Friedel-Anderson Impurities

TL;DR

This paper numerically investigates the fidelity of ground states in Kondo and Friedel-Anderson impurities using FAIR theory, revealing different behaviors related to Anderson orthogonality catastrophe and phase shifts.

Contribution

It introduces a numerical method to calculate fidelities for impurity models and compares their behaviors, highlighting differences in orthogonality catastrophe effects.

Findings

Kondo impurity fidelity diverges with increasing effective states.

Fidelity saturates for symmetric Friedel-Anderson impurity at large N_{eff}.

Fidelity behavior helps determine electron phase shifts.

Abstract

The fidelities of the Kondo and the Friedel-Anderson (FA) impurities are calculated numerically. The ground states of both systems are calculated with the FAIR (Friedel artificially inserted resonance) theory. The ground state in the interacting systems is compared with a nullstate in which the interaction is zero. The different multi-electron states are expressed in terms of Wilson states. The use of N Wilson states simulates the use of a large effective number N_{eff} of states. A plot of ln(F) versus N\proptoln(N_{eff}) reveals whether one has an Anderson orthogonality catastrophe at zero energy. The results are at first glance surprising. The ln(F)-ln(N_{eff}) plot for the Kondo impurity diverges for large N_{eff}. On the other hand, the corresponding plot for the symmetric FA impurity saturates for large N_{eff} when the level spacing at the Fermi level is of the order of the singlet-triplet excitation energy. The behavior of the fidelity allows one to determine the phase shift of the electron states in this regime. PACS: 75.20.Hr, 71.23.An, 71.27.+a, 05.30.-d