Exact results for the Kondo screening cloud of two helical liquids

We analyze the screening of a magnetic quantum dot with spin 1/2 coupled to two helical liquids. Interestingly, we find two qualitatively different sets of Toulouse points, i.e., nontrivial parameters for which we can solve the two channel Kondo model exactly. This enables us to calculate the temperature and voltage dependent Kondo screening cloud, which develops oscillations for an applied spin voltage ${\mu }_s$. Such a spin voltage can be conveniently applied by a charge bias in a four-terminal helical liquid setup.