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

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.

Kondo holes in strongly correlated impurity arrays: RKKY driven Kondo screening and hole-hole interactions

“…If l only accounts for a single site, the model is known as single-impurity Anderson model that was accurately solved using the NRG [33,34] and the Bethe ansatz [35,36] almost 40 years ago. If the number of sites N f = #l > 1 is small and finite, we refer to a multi impurity Anderson model (MIAM), H MIAM , whose simplest realization is the two-impurity Anderson model (TIAM) [37,38].…”

“…In the wide band limit, V i /t → 0, the energy dependence of the hybridization function matrix can be neglected, ∆(ω − i0 + ) ≈ ∆(−i0 + ), and we can absorb the real part into the energy matrix: E → E ′ = E + ℜ∆(−i0 + ). Using the unitary transformation U that diagonalizes the remaining imaginary part, Γ diag = U ℑ∆(−i0 + )U * , the approximated Green's function reads (38) with Ẽ′ = U E ′ U * . Consequently, the single particle Green's function matrix G f (ω − i0 + ), in the eigenbase of ℑ∆(−i0 + ), can equally be generated by an effective single particle Hamiltonian Hsp which has the following form:…”

“…Consequently, the intermediate LM FP disappears and the RG directly crosses over from the high temperature unstable FP that contains Ñf = 6 independent spin-1/2 moments to the IIS FP with singlet ground state. However, the SC and the ISS FP are adiabatically connected such that there is no additional QCP between a Kondo screened singlet and a RKKY driven singlet, just as in the extensively studied TIAM [37,38,55,56,68,69].…”

Magnetic correlation between two local spins in a quantum spin Hall insulator

“…It has been proposed that controllable RKKY interaction can be used to manipulate the quantum states of local spins, which is very helpful for spintronics as well as quantum computing 23,24,30 . Recently, the RKKY interactions in graphene [31][32][33][34][35][36][37] and spin-orbital systems [38][39][40][41][42][43][44][45][46] have been intensely investigated. It has been demonstrated that 33,34 for a honeycomb lattice at half filling, with hopping only between different sublattices, the RKKY interaction is FM for impurities located at the same sublattice and AFM for impurities at the different sublattices.…”

Nonlocal pairing as a source of spin exchange and Kondo screening

“…In a geometry where the two QDs contact the same lead, conduction band electrons mediate the RKKY exchange [11][12][13]. The RKKY interaction competes with the Kondo effect and leads to the quantum phase transition of a still debated nature [14][15][16][17][18][19][20][21][22]. Moreover, in DQDs coupled in series also superexchange can alter the Kondo physics significantly [23,24].…”