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We analyze the screening of a magnetic quantum dot with spin 1/2 coupled totwo helical liquids. Interestingly, we find two qualitatively different sets ofToulouse points, i.e., nontrivial parameters for which we can solve the twochannel Kondo model exactly. This enables us to calculate the temperature andvoltage dependent Kondo screening cloud, which develops oscillations for anapplied spin voltage $\mu_s$. Such a spin voltage can be conveniently appliedby a charge bias in a four-terminal helical liquid setup.

Exact results for the Kondo screening cloud of two helical liquids