  We consider the Kondo effect in Y-junctions of anisotropic XY models in an
applied magnetic field along the critical lines characterized by a gapless
excitation spectrum. We find that, while the boundary interaction Hamiltonian
describing the junction can be recasted in the form of a four-channel, spin-1/2
antiferromagnetic Kondo Hamiltonian, the number of channels effectively
participating in the Kondo effect depends on the chain parameters, as well as
on the boundary couplings at the junction. The system evolves from an effective
four-channel topological Kondo effect for a junction of XX-chains with
symmetric boundary couplings into a two-channel one at a junction of three
quantum critical Ising chains. The effective number of Kondo channels depends
on the properties of the boundary and of the bulk. The XX-line is a "critical"
line, where a four-channel topological Kondo effect can be recovered by
fine-tuning the boundary parameter, while along the line in parameter space
connecting the extreme regimes, XX-line and the critical Ising point the
junction is effectively equivalent to a two-channel topological Kondo
Hamiltonian. Using a renormalization group approach, we determine the flow of
the boundary couplings, which allows us to define and estimate the critical
couplings and Kondo temperatures of the different Kondo (pair) channels.
Finally, we study the local transverse magnetization in the center of the
Y-junction, eventually arguing that it provides an effective tool to monitor
the onset of the two-channel Kondo effect.
