  We propose and study a parafermionic generalization of the topological Kondo
effect. The latter has been predicted to arise for a Coulomb-blockaded
mesoscopic topological superconductor (Majorana box), where at least three
normal leads are tunnel-coupled to different Majorana zero modes on the box.
The Majorana states represent a quantum impurity spin that is partially
screened due to cotunneling processes between leads, with a stable non-Fermi
liquid ground state. Our theory studies a generalization where (i) Majorana
states are replaced by topologically protected parafermionic zero modes, (ii)
charging effects again define a spin-like quantum impurity on the resulting
parafermion box, and (iii) normal leads are substituted by fractional edge
states. In this multi-terminal problem, different fractional edge leads couple
only via the parafermion box. We show that although the linear conductance
tensor exhibits similar behavior as in the Majorana case, both at weak and
strong coupling, our parafermionic generalization is actually not a Kondo
problem but defines a rich new class of quantum impurity problems. At the
strong-coupling fixed point, a current injected through a reference lead will
be isotropically partitioned into outgoing currents in all other leads,
together with a universal negative current scattered into the reference lead.
The device can thus be operated as current extractor, where the current
partitioning is noiseless at the fixed point. We describe a fractional quantum
Hall setup proximitized by superconductors and ferromagnets, which could allow
for an experimental realization in the near future.
