Numerical renormalization group study of two-channel three-impurity triangular clusters

We study triangular clusters of three spin-1/2 Kondo or Anderson impurities that are coupled to two conduction leads. In the case of Kondo impurities, the model takes the form of an antiferromagnetic Heisenberg ring with Kondo-type exchange coupling to continuum electrons. We show that this model exhibits many types of the behavior found in various simpler one- and two-impurity models, thereby enabling the study of crossovers between a number of Fermi-liquid (FL) and non-Fermi-liquid (NFL) fixed points. In particular, we explore a direct crossover between the two-impurity Kondo-model NFL fixed point and the two-channel Kondo-model NFL fixed point. We show that the concept of the two-stage Kondo effect applies even in the case when the first-stage Kondo state is of NFL type. In the case of Anderson impurities, we consider the transport properties of three coupled quantum dots. This class of models includes, as limiting cases, the familiar serial double quantum dot and triple quantum dot nanostructures. By extracting the quasiparticle scattering phase shifts, we compute the low-temperature conductance as a function of the interimpurity tunneling coupling. We point out that due to the existence of exponentially low-temperature scales, there is a parameter range where the stable “zero-temperature” fixed point is essentially never reached (not even in numerical renormalization group calculations). The zero-temperature conductance is then of no interest and it may only be meaningful to compute the conductance at finite temperature. This illustrates the perils of studying the conductance in the ground state and considering thermal fluctuations only as a small correction.

Figure 1

(Color online) Two-channel three-impurity triangular cluster.

Figure 2

(Color online) Schematic representations of the possible fixed points.

Figure 3

(Color online) Schematic phase diagram of the triangular three-impurity Kondo model at low temperatures.

Figure 4

(Color online) NRG eigenvalue renormalization flow diagrams for even iteration numbers $N$. The states are indexed by the total isospin, total spin, and parity quantum numbers.

Figure 5

(Color online) Spin-spin-correlation functions between the impurities in the upper arm of the triangle $\left({\mathit{S}}_1\cdot {\mathit{S}}_3\right)$ and between the side-coupled impurity and one of the impurities in the upper arm $\left({\mathit{S}}_1\cdot {\mathit{S}}_2\right)$. Note that ${\mathit{S}}_1\cdot {\mathit{S}}_2={\mathit{S}}_3\cdot {\mathit{S}}_2$ due to reflection symmetry. The correlations shown are those at $T=0$; in the major part of the parameter space, the final values are established on the scale $\sim \text{min}\left(J_{13},J_{12}\right)$, which is typically much higher than the Kondo temperature scales.

Figure 6

(Color online) Impurity cluster contribution to the magnetic susceptibility and entropy for a range of values of $\beta$ at fixed $J_0$.

Figure 7

(Color online) The Kondo crossover temperature as a function of the parameter $\beta$ for fixed $J_0/D=0.0158$.

Figure 8

(Color online) Impurity cluster contribution to the entropy for a range of values of $J_0$ at fixed $\beta$. $T_{\Delta }^{2\text{IK}}$ is driven down until it is made equal to $T_K^{2\text{CK}}$.

Figure 9

(Color online) NRG flow diagram in the case of the crossover from 2IK to 2CK regimes.

Figure 10

(Color online) Schematic phase diagram in the $\left(T,J\right)$ plane for constant parameter $\beta$.

Figure 11

The crossover (Kondo) temperatures $T_K^{\left(2\text{CK}\right)}$ at which the system approaches the two-channel Kondo-model fixed point as a function of $J_0$ and $\beta$. The numbers shown are the integer parts of the decadic logarithms, $\left|{\text{log}}_{10}\left(T_K^{\left(2\text{CK}\right)}/D\right)\right|$. The omitted values for intermediate values of $J_0$ correspond to the situation where it is difficult to determine $T_{\left(2CK\right)}$ since the crossover occurs at relatively high temperature.

Figure 12

(Color online) Zero-temperature linear conductance through the upper edge of the triangular cluster and the quasiparticle scattering phase shifts as a function of the interdot-tunneling-asymmetry factors $\alpha$ for a range of tunneling parameters $t_0$. In the shaded box, the spectrum at the last NRG iteration does not correspond to that of a Fermi-liquid fixed point. The critical $t^{\ast }$ corresponding to the maximal conduction in the DQD model is approximately $t^{\ast }=0.028U$.

Figure 13

(Color online) Zero-temperature linear conductance through the upper edge of the triangular cluster as a function of the interdot tunneling $t_0$ for a range of tunneling-asymmetry factors $\alpha$.