Kondo temperature of SU(4) symmetric quantum dots

A path integral approach is used to derive a closed analytical expression for the Kondo temperature of the SU(4) symmetrical Anderson model. In contrast to the SU(2) case, the prefactor of the Kondo temperature is found to display a peculiar orbital-energy (gate voltage) dependence, reflecting the presence of various SU(4) mixed valence fixed points. Our analytical expressions are tested against and confirmed by numerical renormalization group computations.

Figure 1

Sketch of the “phase diagram” of the SU(4) Kondo model. In the mixed-valence region ${\mathrm{MV}}_1$ $\left({\mathrm{MV}}_2\right)$ the fourfold-degenerate $q=1$ state competes with the $q=0$ empty orbital state (the sixfold-degenerate $q=2$ state). In the local moment region ${\mathrm{LM}}_1$ the $q=1$ SU(4) spin produces Kondo effect and is screened below the Kondo scale, $T_{\mathrm{K}}$, while in region ${\mathrm{LM}}_2$ a sixfold-degenerate SU(4) spin is screened. EO and SC stand respectively for empty orbital and strong coupling [18].

Figure 2

(left panel) Comparison of the analytical prefactors $f_1\left(x\right)$ and $f_2\left(x\right)$ in Eq. (4) (solid lines) with ${\mathcal{D}}_{\mathrm{eff}}/U$, extracted from NRG calculations (symbols). The numerical calculations were carried out for ${\nu }_0{J}_0=\{0.1,0.25\}$. The sector $q=3$ is symmetric to that with $q=1$: $f_3(x-3)=f_1(-x)$. (right panel) Comparison of $T_{\mathrm{K}}/U$ for $U/\Delta =4$ with the effective cutoff of Eq. (4) (solid line) or a constant prefactor $\sqrt{2U\Delta /\pi }$ instead of $D_{\mathrm{eff}}\sqrt[4]{2{\nu }_0{J}_0}$ in Eq. (3) (dashed line).

Figure 3

Temperature dependence of impurity contribution to the entropy.

Figure 4

Diagrammatic expansion from Eq. (6) for the spin-exchange process $\mathcal{J}\left(\omega \right)$ to second order in $\Delta$. Scattering lead electrons are considered at the energy $\omega$, while slave fermions are considered at energy ${\tilde{\varepsilon }}_d$.