Engineering the localization transition in a Charge-Kondo circuit

Abstract

Charge Kondo circuits consist of metallic islands connected by single-mode quantum point contacts (QPCs). The island's charging energy makes these circuits tunable quantum simulators of various strongly interacting models. Here we propose a circuit that realizes the Kondo effect with effective Luttinger-liquid interactions, and show that it undergoes a localization transition in which the QPC transmission is fully suppressed below a critical value. Experimental signatures include a diverging charge susceptibility and an entropy step. Our findings open a path toward realizing localization transitions in more exotic settings.

Figures (5)

• Figure 1: (a) Charge Kondo circuit with Luttinger-liquid interactions. A resistor is connected to the node (a large metallic dot without charge quantization), and electron tunneling occurs between this node and the Kondo metallic dot. (b) Phase diagram of the anisotropic Kondo model in Eq. \ref{['eq:RG0']}. The red (blue) region corresponds to the localized (delocalized) phase.
• Figure 2: Circuit node as a metallic island with dominant charging energy, serving as a simulator of LL physics lee2020fractionalPhysRevB.105.075433. It connects to $n+1$ integer quantum Hall edge channels, one of which is partitioned by a QPC and coupled to a charge-Kondo island via a weak transmission amplitude $t$.
• Figure 3: NRG results for $n=5$. (a)$\langle\hat{N}\rangle$ versus $N_g$ for decreasing tunnelings. For $t>t_c$ (dark to light green), the charge state is delocalized, and we find a smooth crossover. For $t<t_c$ (dark to light pink), the charge state is localized, and we find a discontinuity at the transition point $N_g=1/2$. (b) Charge-susceptibility versus temperature for decreasing tunnelings. In the delocalized phase, $\chi$ saturates in the low temperature regime, while in the localized phase, $\chi$ keeps increasing down to $T=0$. (c) Left: inverse susceptibility versus $t$ for decreasing temperatures, for lines with colors changing from red to blue. Right: inverse susceptibility versus $1/\sqrt{t-t_c}$, where $t_c=0.3782$, for the same temperatures. In the limit $T\to 0$, the curves exhibit the KT scaling behavior $\chi^{-1}\sim \exp(-\mathrm{const}/\sqrt{t-t_c})$.
• Figure 4: NRG results for the critical tunneling $t_c$ versus $n$, demonstrating that $t_c \propto n^{-1}$ near the KT transition.
• Figure 5: NRG results for the impurity entropy $S_{imp}$, obtained from the Hamiltonian Eq. \ref{['eq:H_tot_large_r']} with $n=5$; see text for further discussion.