We show through a perturbative and an exact calculation using Wilson's renormalization methods that in the problem of two interacting Kondo impurities, on-site potential scattering generates a quantum-tunneling between the two impurities through a marginally relevant operator. The magnitude of this tunneling
V12 depends on the spin-correlation between the two impurities. For exchange interactions between the moments comparable to the Kondo energy
TK,
V12 is typically much larger than
TK. This implies that the heavy-fermion mass in this interesting range is determined by
V12 rather than
TK. The importance of these results for experiments in coupled Kondo dots is also pointed out.
The Kondo problem, which describes the interaction of a spin
s magnetic impurity with a free Fermi gas, is a classic example of strongly coupled physics. Historically, the problem has been solved by Wilson's numerical renormalization group and later by Bethe ansatz. In this paper, we present an alternate analytic solution of the Kondo problem that combines an expansion in
1/s with the renormalization group. We study both the case of an impurity interacting with a single channel
K=1 of fermions in the
s→∞ limit and the case with
K channels in the double-scaling limit
K→∞,
s→∞,
K/s fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large
s and compute thermodynamic observables such as the impurity entropy and susceptibility. We find these observables to agree with the Bethe ansatz results. We also compute the impurity spectral function, finite temperature resistivity and the Kondo screening cloud profile, properties that are not easily accessible from Bethe ansatz. Notably, in the regime
K>2s we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.
We present an extensive study of the two-impurity Kondo problem for spin-1 adatoms on a square lattice using an exact canonical transformation to map the problem onto an effective one-dimensional system that can be numerically solved using the density matrix renormalization group method. We provide a simple intuitive picture and identify the different regimes, depending on the distance between the two impurities, Kondo coupling
JK, longitudinal anisotropy
D, and transverse anisotropy
E. In the isotropic case, two impurities on opposite (the same) sublattices have a singlet (triplet) ground state. However, the energy difference between the triplet ground state and the singlet excited state is very small and we expect an effectively fourfold-degenerate ground state, i.e., two decoupled impurities. For large enough
JK the impurities are practically uncorrelated forming two independent underscreened states with the conduction electrons, a clear nonperturbative effect. When the impurities are entangled in an RKKY-like state, Kondo correlations persist and the two effects coexist: the impurities are underscreened, and the dangling spin-
1/2 degrees of freedom are responsible for the interimpurity entanglement. We analyze the effects of magnetic anisotropy in the development of quasiclassical correlations.
We study the Kondo model---a magnetic impurity coupled to a one-dimensional wire via exchange coupling---by using Wilson's numerical renormalization group technique. By applying an approach similar to which was used to compute the two-impurity problem we managed to improve the poor spatial resolution of the numerical renormalization group method. In this way we have calculated the impurity-spin--conduction-electron-spin correlation function which is a measure of the Kondo compensation cloud whose existence has been a long-standing problem in solid-state physics. We also present results on the temperature dependence of the Kondo correlations.
We propose a model of a strongly-interacting two-impurity Kondo system based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, also known as holography. In a Landau Fermi Liquid, the single-impurity Kondo effect is the screening of an impurity spin at low temperature T . The two-impurity Kondo model then describes the competition between the Kondo interaction and the Heisenberg interaction between two impurity spins, also called the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. For spin-1/2 impurities, that competition leads to a quantum phase transition from a Kondo-screened phase to a phase in which the two impurity spins screen one another. Our holographic model is based on a (1 + 1)-dimensional CFT description of the two-impurity Kondo model, reliable for two impurities with negligible separation in space. We consider only impurity spins in a totally anti-symmetric representation of an SU(N ) spin symmetry. We employ a large-N limit, in which both Kondo and RKKY couplings are double-trace, and both Kondo and inter-impurity screening appear as condensation of single-trace operators at the impurities’ location. We perform the holographic renormalization of our model, which allows us to identify the Kondo and RKKY couplings as boundary conditions on fields in AdS. We numerically compute the phase diagram of our model in the plane of RKKY coupling versus T , finding evidence for a quantum phase transition from a trivial phase, with neither Kondo nor inter-impurity screening, to a non-trivial phase, with both Kondo and anti-ferromagnetic inter-impurity screening. More generally we show, just using SU(N ) representation theory, that ferromagnetic correlations must be absent at leading order in the large-N limit. Our holographic model may be useful for studying many open problems involving strongly-interacting quantum impurities, including for example the Kondo lattice, relevant for describing the heavy fermion compounds.