SO(5) Non-Fermi Liquid in a Coulomb Box Device

Non-Fermi liquid (NFL) physics can be realized in quantum dot devices where competing interactions frustrate the exact screening of dot spin or charge degrees of freedom. We show that a standard nanodevice architecture, involving a dot coupled to both a quantum box and metallic leads, can host an exotic SO(5) symmetry Kondo effect, with entangled dot and box charge and spin. This NFL state is surprisingly robust to breaking channel and spin symmetry, but destabilized by particle-hole asymmetry. By tuning gate voltages, the SO(5) state evolves continuously to a spin and then “flavor” two-channel Kondo state. The expected experimental conductance signatures are highlighted.

Figure 1

Right: schematic of the device. A quantum dot coupled to a quantum box and source and drain leads. Left: NRG phase diagram spanned by dot and box gate voltages, $V_d\propto \eta /U_d$ and $V_B\propto n_B$, showing the NFL line for various channel asymmetries $t_L/t_B$. SO(5) point located at $n_B=\pm \frac{1}{2}$ and $\eta =0$. Plotted for constant $U_d=0.3$, $E_C=0.1$, and $t_B=0.12$.

Figure 2

Physical properties of the SO(5) Kondo effect, obtained by NRG for $n_B=\frac{1}{2}$, $\eta =0$, $U_d=0.3$, $E_C=0.1$, $t_L=0.085$, and $t_B\simeq \xi t_L=0.1$. (a) Impurity contribution to entropy $S_{\mathrm{imp}}(T)$, showing partial quenching of the entangled spin and flavor degrees of freedom on the scale of the Kondo temperature $T_K\sim {10}^{-4}$. For $T_K\ll T\ll E_C$, free impurity spin and flavor give a ln(4) entropy, while $S_{\mathrm{imp}}(T)=\frac{1}{2}\ln (2)$ for $T\ll T_K$, characteristic of the free Majorana fermion at the SO(5) fixed point. (b) $T=0$ local spin and flavor dynamical susceptibilities, both showing apparent FL-like behavior ${\chi }_{\mathrm{loc}}(\omega )\sim \omega$ for $\omega \ll T_K$. (c) Linear response conductance through the dot $G(T)/G_0$ (blue line), with $G=\frac{1}{2}{G}_0$ at $T=0$, and leading behavior $G(T)-G(0)\sim +(T/T_K{)}^{3/2}{G}_0$ (inset, dashed line). The standard spin-2CK conductance line shape is given for comparison as the dotted line.

Figure 3

Evolution along the black NFL line in Fig. 1. (a) $T=\omega =0$ susceptibilities ${\chi }_{\mathrm{loc}}^s$ (black), ${\chi }_{\mathrm{loc}}^f$ (red), and Kondo temperature $T_K$ (blue). Flavor fluctuations are suppressed in the box Coulomb blockade regime, while spin fluctuations are suppressed as the dot local moment is lost for $n_B\to -\frac{1}{2}$. (b) Behavior near $n_B=\frac{1}{2}$ showing ${\chi }_{\mathrm{loc}}^s(0)\sim (\frac{1}{2}-n_B{)}^2$ [consistent with ${\chi }_{\mathrm{loc}}^s(\omega )\sim \omega$], while ${\chi }_{\mathrm{loc}}^f$ remains finite. (c) $T_K{\chi }_{\mathrm{loc}}^s$ (black dashed) and $T_K{\chi }_{\mathrm{loc}}^f$ (red dashed) showing crossover from spin-flavor to S-2CK to F-2CK as $n_B$ is varied from $+\frac{1}{2}$ to $-\frac{1}{2}$.