Tunable Kondo screening length at a Y-junction of three inhomogenous spin chains

We derive the topological Kondo Hamiltonian describing a Y junction of three XX-spin chains connected to outer quantum Ising chains with different tilting angles for the Ising axis. We show that the tilting angles in the spin models play the role of the phases of the superconducting order parameters at the interfaces between bulk superconductors and one-dimensional conducting normal electronic wires. As a result, different tilting angles induce nonzero equilibrium spin (super)currents through the junction. Employing the renormalization group approach to the topological Kondo model, we derive the scaling formulas for the equilibrium spin currents. We argue that, by monitoring the crossover in the currents induced by the Kondo effect, it is possible to estimate the Kondo screening length. In particular, we prove how it is possible to tune the Kondo length by acting on the applied phases only; this enables us to map out the scaling properties by just tuning the tilting angles and the Kondo length accordingly.