Quantum phase transition in quantum dot trimers

We investigate a system of three tunnel-coupled semiconductor quantum dots in a triangular geometry, one of which is connected to a metallic lead, in the regime where each dot is essentially singly occupied. Both ferromagnetic and antiferromagnetic spin-$\frac{1}{2}$ Kondo regimes, separated by a quantum phase transition, are shown to arise on tuning the interdot tunnel couplings and should be accessible experimentally. Even in the ferromagnetically-coupled local moment phase, the Kondo effect emerges in the vicinity of the transition at finite temperatures. Physical arguments and numerical renormalization group techniques are used to obtain a detailed understanding of the problem.

Figure 1

Schematic illustration of the quantum dot trimer.

Figure 2

Entropy $S_{\text{imp}}\left(T\right)$ vs $T/\Gamma$ for fixed $\tilde{U}=10$ and $\tilde{t}=0.1$, as a function of ${\tilde{t}}^{\prime }$ as the transition (${\tilde{t}}_c^{\prime }\simeq 0.097$) is approached from either side: for $t^{\prime }=t_c^{\prime }\pm \lambda T_K$ with $\lambda ={10}^4,10,1,{10}^{-1},{10}^{-4}$, and $\simeq 0$ [curves (a)–(f), respectively]. Solid lines: antiferromagnetically coupled SC phase $\left(\lambda >0\right)$; dashed lines: ferromagnetically coupled LM phase $\left(\lambda <0\right)$. Inset: close to the transition, the scale $T_{\Delta }$ vanishes linearly in $\left({\tilde{t}}^{\prime 2}-t_c^{\prime 2}\right)$.

Figure 3

Spin susceptibility $T{\chi }_{\text{imp}}\left(T\right)$ vs $T/\Gamma$ for the same parameters as Fig. 2. Solid lines: antiferromagnetically coupled SC phase; dashed lines: ferromagnetically coupled LM phase. Close to the transition, for $\left|{\tilde{E}}_{\Delta }\right|<T<T_K$, a $T{\chi }_{\text{imp}}\left(T\right)=\frac{1}{6}$ plateau symptomatic of the transition fixed point arises, persisting down to $T=0$ precisely at the transition where ${\tilde{E}}_{\Delta }=0$ [case (f)].

Figure 4

$T=0$ local spectrum $\pi \Gamma D_2\left(\omega ;T=0\right)$ vs $\omega /\Gamma$ for systems with the same parameters as Fig. 2. Panels (A) and (B) show all data on logarithmic and linear frequency scales, respectively, with solid lines again for systems in the SC phase $\left(t^{\prime }>t_c^{\prime }\right)$ and dashed lines for the LM phase $\left(t^{\prime }<t_c^{\prime }\right)$. Inset (C) shows the low-frequency behavior in the LM phase, fitted to a line of the form $\pi \Gamma D_2\left(\omega ;T=0\right)=a/{\text{ln}}^2\left(\left|\omega \right|/T_0\right)$.

Figure 5

Conductance $G\left(T\right)/G_0$ vs $T/T_K$ for the same parameters as Fig. 2. As $T\to 0$, all systems in the SC phase (solid lines) satisfy the unitarity limit $G\left(T=0\right)/G_0=1$ while in the LM phase (dashed lines) $G\left(T=0\right)/G_0=0$. When $T_{\Delta }<T<T_K$, the conductance is controlled by the transition FP, with $G\left(T\right)/G_0=\frac{1}{3}$ in both phases.