Zero-Field Kondo Splitting and Quantum-Critical Transition in Double Quantum Dots

Double quantum dots offer unique possibilities for the study of many-body correlations. A system containing one Kondo dot and one effectively noninteracting dot maps onto a single-impurity Anderson model with a structured (nonconstant) density of states. Numerical renormalization-group calculations show that, while band filtering through the resonant dot splits the Kondo resonance, the singlet ground state is robust. The system can also be continuously tuned to create a pseudogapped density of states and access a quantum-critical point separating Kondo and non-Kondo phases.

Figure 1

Schematic of the DQD system. Dot 1 is Kondo-like ($-{\epsilon }_1$, ${\epsilon }_1+U_1\gg {\Delta }_1=\pi {\rho }_0{V}_1^2$, where ${\rho }_0$ is the lead DOS), while dot 2 can be treated as a single, noninteracting ($U_2\ll {\Delta }_2=\pi {\rho }_0{V}_2^2$) resonant level.

Figure 2

Side-dot configuration. (a)–(c) Dot-1 spectral density for $U_1=-2{\epsilon }_1=0.5$, ${\epsilon }_2=0$, and ${\Delta }_2=0.02$. Splitting appears in $A_{11}(\omega )$ for larger $\lambda$ such that $T_K\gtrsim {\Delta }_2/\sqrt{2}$. (d) Effective moment ${\mu }_1^2$ vs $T$. (e) $T_K$ vs $\lambda$. For large $\lambda$, $T_K\propto \lambda$ (dashed curve). (Energies and $T_K$ in units of $D$.)

Figure 3

Parallel DQD with $U_1=0.5$, ${\Delta }_1=0.05$, ${\Delta }_2=0.02$, and ${\epsilon }_2=0$. (a) Hybridization $\Delta (\omega )$ vanishes as ${\omega }^2$ at $\omega =0$. (b)–(d) Dot-1 spectral density $A_{11}(\omega )$ and (e) effective moment ${\mu }_1^2(T)$, for various ${\epsilon }_1$. For ${\epsilon }_1\approx -U_1/2$, no Kondo effect occurs: $A_{11}(0)=0$ in (b) and ${\mu }_1^2(0)=1/4$ [▪ and □ in (e)]. This LM phase is separated by a QCP at ${\epsilon }_1={\epsilon }^{*}>-U_1/2$, characterized by a featureless $A_{11}(\omega )$ in (c) and ${\mu }_1^2(0)=1/6$ [● in (e)], from the SC phase [(d), ▴ and ▵ in (e)] in which $A_{11}$ vanishes at $\omega =0$ and peaks at some $\omega >0$.

Figure 4

Fully connected DQD with $U_1=-2{\epsilon }_1=0.5$, ${\Delta }_1={\Delta }_2=0.02$, and $\lambda =0.1$. (a)–(e) Hybridization $\Delta (\omega )$ for various ${\epsilon }_2$ and (f)–(j) the corresponding $A_{11}(\omega )$. An asymmetric Kondo peak appears when $\Delta (\omega )$ is featureless near $\omega =0$ (a),(e). The peak splits as the resonance in $\Delta (\omega )$ approaches $\omega =0$ (b), reappears in the intermediate region (c), and disappears altogether when a pseudogap forms (d).