The 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 multichannel 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 (RG) approach. We recover the results for the `decoupled' limit (interactions between the helical liquids switched off) and point out deviations from the known results due to this coupling. The perturbative RG study shows that each of the channels can flow to either a Ferromagnetic (FM) or an Anti-Ferromagnetic (AFM) fixed point. We obtain the phase diagram of the coupled system as a function of the system parameters. Observable consequences of the coupling are captured using linear response theory, which shows the scaling of the temperature as a function of the inter-channel interaction parameters.