Quantum Phase Transition in Capacitively Coupled Double Quantum Dots

We investigate two equivalent, capacitively coupled semiconducting quantum dots, each coupled to its own lead, in a regime where there are two electrons on the double dot. With increasing interdot coupling, a rich range of behavior is uncovered: first a crossover from spin- to charge-Kondo physics, via an intermediate $SU(4)$ state with entangled spin and charge degrees of freedom, followed by a quantum phase transition of Kosterlitz-Thouless type to a non-Fermi-liquid “charge-ordered” phase with finite residual entropy and anomalous transport properties. Physical arguments and numerical renormalization group methods are employed to obtain a detailed understanding of the problem.

Figure 1

(a) Phase diagram in the $({\tilde{U}}^{\prime },\tilde{U})$ plane; the $SU(4)$ line $U^{\prime }=U$ is also shown (dotted line). (b) For $\tilde{U}=7$ in the SC phase, $\ln (T_K/\Gamma )$ vs $[{\tilde{U}}_c^{\prime }-{\tilde{U}}^{\prime }{]}^{-1/2}$ close to the transition, showing the exponential vanishing of $T_K$. Inset: $\ln (T_K/\Gamma )$ vs ${\tilde{U}}^{\prime }$.

Figure 2

$S(T)$ vs $T/\Gamma$ for $\tilde{U}=7$ (${\tilde{U}}_c^{\prime }\simeq 7.046$). (a) ${\tilde{U}}^{\prime }=6$ (SC, “uncoupled spin-Kondo”) and $8$ (CO); (b) ${\tilde{U}}^{\prime }=6.9$ [crossover to $SU(4)$]; (c) ${\tilde{U}}^{\prime }=7$ [$SU(4)$]; (d) ${\tilde{U}}^{\prime }=7.03$ (charge-Kondo). Dotted lines show $\ln 16$, $\ln 6$, $\ln 4$, and $\ln 2$, associated with the FO, ${\mathrm{LM}}^{SU(4)}$, ${\mathrm{LM}}^{SU(2)}$, and CO fixed points, respectively.

Figure 3

Transmission $\pi \Gamma D_i(\omega )$ vs $\omega /\Gamma$, for $\tilde{U}=7$ and (top to bottom) ${\tilde{U}}^{\prime }=7,7.03,7.044$ (SC) and $7.048,7.1$ (CO).