Quantum spins on star graphs and the Kondo model

We study the XX model for quantum spins on the star graph with three legs (i.e., on a Y  -junction). By performing a Jordan–Wigner transformation supplemented by the introduction of an auxiliary space we find a Kondo Hamiltonian of fermions, in the spin 1 representation of su(2)$\mathit{su}(2)$, locally coupled with a magnetic impurity. In the continuum limit our model is shown to be equivalent to the 4-channel Kondo model coupling spin-1/2 fermions with a spin-1/2 impurity and exhibiting a non-Fermi liquid behavior. We also show that it is possible to find an XY model such that – after the Jordan–Wigner transformation – one obtains a quadratic fermionic Hamiltonian directly diagonalizable.