Solution of the two-impurity Kondo model: Critical point, Fermi-liquid phase, and crossover

Abstract

An asymptotically exact solution is presented for the two-impurity Kondo model for a finite region of the parameter space surrounding the critical point. This region is located in the most interesting intermediate regime where RKKY interaction is comparable to the Kondo temperature. After several exact simplifications involving reduction to one dimension and abelian bosonization, the critical point is explicitly identified, making clear its physical origin. By using controlled low energy projection, an effective Hamiltonian is mapped out for the finite region in the phase diagram around the critical point. The completeness of the effective Hamiltonian is rigorously proved from general symmetry considerations. The effective Hamiltonian is solved exactly not only at the critical point but also for the surrounding Fermi-liquid phase. Analytic crossover functions from the critical to Fermi-liquid behavior are derived for the specific heat and staggered susceptibility. It is shown that applying a uniform magnetic field has negligible effect on the critical behavior. A detailed comparison is made with the numerical renormalization group * Current address:

Convergence of Energy Scales on the Approach to a Local Quantum Critical Point

“…2, the corresponding values ofJ=Á are shown and all converge to the limiting valueJ=Á ¼ 2, which confirms the predictions given in Eq. (8). The results also apply for the model with U 12 Þ 0 as it is found that…”

“…We have shown in earlier work how the renormalized parameters can be deduced from an analysis of the low energy fixed point of an NRG calculation [16,19], and we apply the same procedure for this model to calculate the low energy behavior and test the predictions given in Eqs. (8) and (9). In Fig.…”

“…It is difficult to test this conjecture for a lattice model of this situation using the theoretical techniques currently available. There is, however, a twoimpurity Kondo model which has been shown to have a quantum critical point [5][6][7][8][9], which might throw some further light on this particular mechanism. In this Letter, we study a related two-impurity model using a combination of renormalized perturbation theory (RPT) and numerical renormalization group (NRG) calculations.…”

Phase diagram and critical points of a double quantum dot

“…It can describe a single magnetic impurity with two fold 'orbital' degeneracy, where the exchange coupling J then corresponds to a Hund's rule term where a ferromagnetic exchange term would be appropriate [11][12][13] . In the strong coupling regime U/π∆ ≫ 1, the model corresponds to two Kondo models and with U 12 = 0 it has been much studied as a model of a system with two coupled magnetic impurities 7,[14][15][16] . The model with J = 0 has also been used to study two capacitatively coupled quantum dots 9 .…”

Kondo Impurities Coupled to a Helical Luttinger Liquid: RKKY-Kondo Physics Revisited

“…If the temperature is below the Kondo temperature, T < T K , the antiferromagnetic Kondo coupling drives the single KI to the strong coupling limit where the electrons screen KI. Hence, the Kondo screening is an antagonist of the RKKY interaction.The RKKY-Kondo interplay has attracted a large attention since several decades [3][4][5][6][7] and remains a hot topic of research because of its importance for new systems, such as graphene [8,9], strongly correlated quantum wires and carbon nanotubes, which are described by the Luttinger Liquid model [10,11]. The latter are especially interesting because the Kondo effect can be enhanced by the interactions [12][13][14].…”