Generalized Schrieffer-Wolff Transformation of 2 Kondo Impurity Problem

The low-energy interactions of a system of magneticimpurities coupled to a sea of conduction electrons dis-play a rich variety of physical phenomena. One of themost well-known many-body effects is screening of thelocal moment by the conduction electrons, first explainedby Jun Kondo [1]. The Kondo effect was first observed inmeasurements of dilute magnetic impurities in metals [2],and is now ubiquitious in many systems including quan-tum dots (QD) [3], [4] and magnetic atoms in surfacesprobed by scanning tunneling microscopy [5].In materials with a dense lattice of magnetic impuri-ties, there is an interesting interplay between the Kondoeffect and magnetic ordering of the local moments, whichis of particular interest in the heavy-fermion compounds.As the simplest manifestation of this competition, andas a first step towards understanding the lattice problem,considerable attention has been placed on the two Kondoimpurity problem. The problem was first solved exactlyin the particle-hole symmetric case using NRG [6] andsubsequently by the Bethe ansatz, conformal field theory[7], and bosonization [8]. A second-orderphase transitionisfound for the caseofparticle-holesymmetry, which typ-ically turns into a cross-over when potential scattering isintroduced.A recent two quantum dot experiment by Craig et al.[9] in the geometry shown in Fig. 1 showed a changein the Kondo resonance of one dot by tuning the sec-ond dot from an even to an odd occupation. It has beensuggested that this is due to RKKY coupling betweenthe dots. We have looked at the effective inter-dot in-teractions obtained via a canonical transformation of thetwo impurity Anderson model. The purpose of the cal-culation is to extract the low-energy spin excitations ofthe system in order to predict the magnetic interactions.We expect such interactions to occur in STM systems ofmagnetic impurities on surfaces as well.