Multiple-impurity Anderson model for quantum dots coupled in parallel

The system of several $\left(N\right)$ quantum dots coupled in parallel to the same single-mode conduction channel can be modeled as a single-channel $N$-impurity Anderson model. Using the generalized Schrieffer-Wolff transformation we show that near the particle-hole symmetric point, the effective Hamiltonian in the local moment regime is the $N$-impurity $S=1∕2$ Kondo model. The conduction-band-mediated RKKY exchange interaction between the dots is ferromagnetic and at intermediate temperatures locks the moments into a maximal spin $S=N∕2$ ground state. We provide an analytical estimate for the RKKY interaction. At low temperatures the spin is partially screened by the conduction electrons to $N∕2-1∕2$ due to the Kondo effect. By comparing accurate numerical renormalization group results for magnetic susceptibility of the $N$-impurity Anderson model to the exact Bethe ansatz results of a $S=N∕2$ SU(2) Kondo system we show that at low-temperature the quantum dots can be described by the effective $S=N∕2$ Kondo model. Moreover, the Kondo temperature is independent of the number of impurities $N$. We demonstrate the robustness of the spin $N∕2$ ground state as well as of the associated $S=N∕2$ Kondo effect by studying the stability of the system with respect to various experimentally relevant perturbations. We finally explore various quantum phase transitions driven by these perturbations.

Figure 1

(Color online) Systems of parallel quantum dots. The tight-binding hopping parameter $t$ determines the half-width of the conduction band, $D=2t$, while parameter $t^{\prime }$ is related to the hybridization $\Gamma$ by $\Gamma ∕D={(t^{\prime }∕t)}^2$.

Figure 2

(Color online) (a) Temperature-dependent susceptibility and (b) entropy of the $N$-dot systems calculated using the NRG. The symbols in the susceptibility plots were calculated using the thermodynamic Bethe ansatz approach for the corresponding $S=N∕2$ SU(2) Kondo models (●: $S=1∕2$, ∎: $S=1$, ◆: $S=3∕2$, and ▴: $S=2$).

Figure 3

(Color online) Temperature-dependence of susceptibility, charge fluctuations $⟨{\left(\delta n\right)}^2⟩$, total spin $S$, and the spin-spin correlations $⟨{\mathit{S}}_1\bullet {\mathit{S}}_2⟩$ of the 2-dot system.

Figure 4

(Color online) (a) Temperature-dependent susceptibility of the 2- and 3-dot systems with the same $\Gamma ∕U$ ratio. Open (closed) symbols are Bethe ansatz results for the $S=1$ $(S=3∕2)$ Kondo model. (b) Comparison of LM-FF transition temperature $T_F^{*}$ with predictions of the perturbation theory. (c) Comparison of calculated $T_K$ with Haldane’s formula.

Figure 5

(Color online) (a) Temperature-dependent susceptibility of the 2-dot system for equal $e\text{−}e$ repulsion $U∕D=0.01$ and for different hybridization strengths $\Gamma$. Symbols represent the Bethe ansatz susceptibility for the $S=1$ Kondo model with corresponding $T_K$. (b) Comparison of calculated and predicted $T_F^{*}$. (c) Fit of $T_K$ to Haldane’s formula, Eq. (5).

Figure 6

(Color online) (a) Temperature-dependent susceptibility of the 2-dot systems on departure $(\delta >0)$ from the particle-hole symmetric point, $\delta =0$. Symbols are fits to the universal susceptibility obtained using the Bethe ansatz method for the $S=1$ Kondo model. (b) Calculated and predicted Kondo temperature, Eq. (35). For comparison we also plot $T_K$ given by Eq. (5), which shows expected discrepancy for large $\delta ∕U$. (c) Calculated $T_F^{*}$ and the fit to an exponential function.

Figure 7

(Color online) (a) Temperature-dependent susceptibility of the 2-dot system with unequal (detuned) on-site energies, ${\delta }_1=\Delta$, ${\delta }_2=-\Delta$. Full symbols present Bethe ansatz results of the equivalent $S=1$ Kondo model, while empty symbols are BA results of a $S=1∕2$ Kondo model. (b) Comparison of calculated and predicted Kondo temperature, see Eqs. (5, 38). (c) The Kondo temperature of the $S=1∕2$ Kondo screening on the singlet side of the transition and a fit to Eq. (39).

Figure 8

(Color online) Temperature-dependent susceptibility of the 2-dot systems for different antiferromagnetic interimpurity couplings $J_{12}$. Circles are Bethe ansatz results for the susceptibility of the $S=1$ Kondo model with the Kondo temperature which is equal for all cases where $J_{12}<J_c$.

Figure 9

(Color online) Temperature-dependent susceptibility of the 2-dot systems with interdot tunneling coupling $t_{12}$. For $t_{12}∕D\approx 2\times {10}^{-4}$, we have $J_{\mathrm{AFM}}∕D\approx 1.6\times {10}^{-5}$, which agrees well with the critical value of $J_c∕D\approx 1.7\times {10}^{-5}$ found in the case of an explicit exchange interaction between the dots, see Fig. 8.

Figure 10

(Color online) Temperature-dependent susceptibility of the 2-dot systems for different interimpurity electron-electron repulsion parameters $U_{12}$. Circles are the Bethe ansatz results for the $S=1∕2$ Kondo model which fit the NRG results in the special case $U_{12}=U$.

Figure 11

(Color online) Temperature-dependent entropy of the 2-dot systems for different interimpurity electron-electron repulsion $U_{12}$.

Figure 12

(Color online) Temperature-dependent susceptibility of the 2-dot system with two-electron hopping between the dots $T_{12}$.

Figure 13

(Color online) Temperature-dependent susceptibility of the 2-dot system with unequal coupling to the leads, ${\Gamma }_2={\alpha }^2{\Gamma }_1$. (a) The range of $\alpha$ where $T_K$ is decreasing. (b) The range of $\alpha$ where $T_K$ is increasing again. Circles (squares) are BA results for the $S=1$ $(S=1∕2)$ Kondo model. The arrows indicate the evolution of the susceptibility curves as the parameter $\alpha$ decreases.

Figure 14

(Color online) Comparison of calculated and predicted Kondo temperature $T_K$ and effective exchange interaction $J_{\mathrm{RKKY}}^{\mathrm{eff}}$. The calculation of scaling results for $T_K$ is described in Appendix .

Figure 15

(Color online) The prefactor $c$ in the RKKY exchange constant $J_{\mathrm{RKKY}}=c16V^4∕\left(D^{2}U\right)$ for a flatband with ${\rho }_0=1∕\left(2D\right)$ as a function of $U$ for a range of values of the impurity energy level $\delta$.

Figure 16

(Color online) Ratio $\xi \left(E\right)=J_{\mathrm{RKKY}}\left(E\right)∕J_{\mathrm{RKKY}}^0$ of the running RKKY coupling constant at energy $E$ over its value in the $E\to 0$ limit. The dashed line is an approximate fit to a simple rational function $\xi \left(E\right)=1∕(1+xE∕U)$.