Two-channel Kondo effect in coupled interacting helical liquids

We study the two-channel Kondo effect in the context of two interacting helical liquids coupled to a spin-$frac12$ magnetic impurity. We show that interactions between the two helical liquids significantly affect the phase diagram and other observable properties. Using a multi-channel Luttinger liquid formalism, we analyze both the Toulouse limit, where an exact solution is available and the weak coupling limit, which can be studied via a perturbative renormalization group approach. We show that contrary to the case of uncoupled liquids, where isotropic couplings lead to non-Fermi liquid behavior and asymmetric couplings lead to Fermi liquid behavior, here the renormalization of the couplings due to the inter-channel interactions allows us to access the non-Fermi liquid fixed point even for asymmetric couplings. By the same token, the Fermi liquid phase can be observed for symmetric couplings.