Effective low-energy description of the two impurity Anderson model: RKKY interaction and quantum criticality

TL;DR

This paper develops an effective low-energy framework for the two-impurity Anderson model, revealing how RKKY interactions include ferromagnetic and anti-ferromagnetic parts, and demonstrates how to restore quantum criticality through an effective tunneling term.

Contribution

It introduces an analytical method to incorporate an effective tunneling term that captures the anti-ferromagnetic RKKY contribution and restores the quantum critical point in the model.

Findings

The anti-ferromagnetic RKKY component can be modeled by an effective tunneling term.

Replacing hybridization functions with symmetric parts and tunneling reproduces the low-temperature spectrum.

Restoring the quantum critical point is possible even without particle-hole symmetry.

Abstract

We show that the RKKY interaction in the two-impurity Anderson model comprise two contributions: a ferromagnetic part stemming from the symmetrized hybridization functions and an anti-ferromagnetic part. We demonstrate that this anti-ferromagnetic contribution can also be generated by an effective local tunneling term between the two impurities. This tunneling can be analytically calculated for particle-hole symmetric impurities. Replacing the full hybridization functions by the symmetric part and this tunneling term leads to the identical low-temperature fixed point spectrum in the numerical renormalization group. Compensating this tunneling term is used to restore the Varma-Jones quantum critical point between a strong coupling phase and a local singlet phase even in the absence of particle-hole symmetry in the hybridization functions. We analytically investigate the spatial frequencies of the effective tunneling term based on the combination of the band dispersion and the shape of the Fermi surface. Numerical renormalization group calculations provide a comparison of the distance dependent tunneling term and the local spin-spin correlation function. Derivations between the spatial dependency of the full spin-spin correlation function and the textbook RKKY interaction are reported.