Conditions for observing emergent SU(4) symmetry in a double quantum dot

We analyze conditions for the observation of a low energy SU(4) fixed point in capacitively coupled quantum dots. One problem, due to dots with different couplings to their baths, has been considered by Tosi, Roura-Bas and Aligia (2015). They showed how symmetry can be effectively restored via the adjustment of individual gates voltages, but they make the assumption of infinite on-dot and inter-dot interaction strengths. A related problem is the difference in the magnitudes between the on-dot and interdot strengths for capacitively coupled quantum dots. Here we examine both factors, based on a two site Anderson model, using the numerical renormalization group to calculate the local spectral densities on the dots and the renormalized parameters that specify the low energy fixed point. Our results support the conclusions of Tosi et al. that low energy SU(4) symmetry can be restored, but asymptotically achieved only if the inter-dot interaction $U_{12}$ is greater than or of the order of the band width of the coupled conduction bath $D$, which might be difficult to achieve experimentally. By comparing the SU(4) Kondo results for a total dot occupation $n_{\rm tot}=1$ and $n_{\rm tot}=2$ we conclude that the temperature dependence of the conductance is largely determined by the constraints of the Friedel sum rule rather than the SU(4) symmetry and suggest that an initial increase of the conductance with temperature is a distinguishing characteristic feature of an $n_{\rm tot}=1$ universal SU(4) fixed point.