Dispersive Impurities in one-dimensional Fermi Gases: From one to two Channel Kondo Polarons

We consider the problem of a dispersive magnetic impurity interacting antiferromagnetically with a one dimensional fermionic gas. By combining general considerations and extensive numerical simulations we show that the problem displays a quantum phase transition between two-channel and one-channel Kondo behaviour upon increasing the Kondo coupling and construct a phase diagramme. We also discuss possible experimental realisations.