Zero-field splitting of the Kondo resonance and quantum criticality in triple quantum dots

We consider a triple-quantum-dot (TQD) system composed by an interacting quantum dot connected to two effectively noninteracting dots, which in turn are both connected in parallel to metallic leads. As we show, this system can be mapped onto a single-impurity Anderson model with a nontrivial density of states. The TQD's transport properties are investigated under a continuous tuning of the noninteracting dots' energy levels, employing the numerical renormalization group technique. Interference between single and many-particle resonances splits the Kondo peak, fulfilling a generalized Friedel sum rule. In addition, a particular configuration in which one of the noninteracting dots is held out of resonance with the leads allows us to access a pseudogap regime where a Kosterlitz-Thouless-type quantum-phase transition (QPT) occurs, separating the Kondo and non-Kondo behavior. Within this same configuration, the TQD exhibits traces of the Fano-Kondo effect, which is in turn strongly affected by the QPT. Signatures of all these phenomena are neatly displayed by the calculated linear conductance.

Figure 1

(a) Schematic of the parallel triple quantum dot setup. Dots 2 and 3 are considered effectively noninteracting single particle levels, while dot 1 is deeply in the Kondo regime. (b) Illustrative representation of the effective density of states $\mathrm{\Delta }\left(\omega \right)$ in the one impurity model.

Figure 2

Zero-temperature conductance vs ${\varepsilon }_3$ for ${\varepsilon }_2=0$ and several values of $t_3/D$: 0.02 (black $•$), 0.03 (red $\times$), 0.04 (brown $⧫$), 0.05 (green $▴$), 0.06 (blue $\blacksquare$). For $t_3=t_2$ (black $•$), $g=0$ regardless ${\varepsilon }_3$. However, when $t_3\ne t_2$, a proper adjustment of ${\varepsilon }_3$ leads to a finite conductance, due to a splitting in the dot-1 ASR.

Figure 3

Zero-temperature spectral function for (a) ${\varepsilon }_3=0.005$ and (b) ${\varepsilon }_3=0.0008$. Here $t_3=0.04D$ and the rest of the parameters are as in Fig. 2. The insets in each panel are amplifications of the respective spectral function near the Fermi energy. As shown in (b), the spectral function nearly vanishes at $\omega =0$ even in the presence of the Kondo effect, in order to fulfill a generalized Friedel sum rule.

Figure 4

Spin susceptibility as a function of temperature for various ${\varepsilon }_3$ (increasing in the direction of the arrow) in the range $[0.001,0.07]$. Here $t_3=0.04D$ and the rest of the parameters are as in Fig. 2. In all curves, $T{\chi }_{\text{imp}}$ vanishes with the temperature, congruent with Kondo physics.

Figure 5

(a) Phase $\varphi$ and (b) $g_0[1-{sin}^2(\pi \langle n_1\rangle /2+\varphi )]$ vs ${\varepsilon }_3$, for ${\varepsilon }_2=0$ and several values of $t_3/D$: 0.03 (red $\times$), 0.04 (brown $⧫$), 0.05 (green $▴$), 0.06 (blue $\blacksquare$). This result shows that the splitting of the ASR can be attributable to a fine tuning of the phase $\varphi$.

Figure 6

(a) Kondo temperature $T_K$ and (b) finite-$T$ conductance $g$ vs ${\varepsilon }_3$. The horizontal lines in (a) correspond to the temperatures in (b) at which the conductance was calculated: $T=0$ (purple $⧫$), $T_1$ (blue $•$), $T_2$ (green $\blacksquare$), $T_3$ (orange $\times$), and $T_4$ (red $▴$). Here $t_3=0.04D$ and the rest of the parameters are as in Fig. 2. As shown by the plots, the phase $\varphi$ can be experimentally determined through electrical conductance measurements.

Figure 7

(a) $T_K$ vs ${\varepsilon }_3$, for several values of $t_3/D$: 0.02 (black $•$), 0.03 (red $\times$), 0.04 (brown $⧫$), 0.05 (green $▴$), 0.06 (blue $\blacksquare$). (b) $ln{T}_K$ vs $-|{\varepsilon }^{*}-{\varepsilon }_3{|}^{-1}$, for $t_3/D=0.04$. In both figures, ${\varepsilon }_2=-0.03D$. The behavior of $T_K$ against ${\varepsilon }_3$ is characteristic of a Kosterlitz-Thouless quantum phase transition.

Figure 8

(a) $g$ vs ${\varepsilon }_3$, for ${\varepsilon }_2/D=-0.03$. Phase $\varphi$ vs ${\varepsilon }_3$ for (b) ${\varepsilon }_2/D=-0.03$ and (c) ${\varepsilon }_2/D=0.03$. In all three panels, we have several values of $t_3/D$: 0.02 (black $•$), 0.04 (brown $⧫$), 0.05 (green $▴$), 0.06 (blue $\blacksquare$). The drastic drop in $g$ at ${\varepsilon }_3>0$ is a result of the quantum phase transition.