Spin and thermal current scaling at a $Y$-junction of XX spin chains

We study the boundary phase diagram and the low-temperature heat and magnetization transport at a $Y$-junction of XX spin chains. Depending on the magnetization axis anisotropy between the magnetic exchange interactions at the junction, the system exhibits two different strong-coupling regimes at low energies/temperatures, similar to the overscreened (topological) four- and to the two-channel Kondo fixed points. Using renormalization group arguments combined with boundary conformal field theory methods, we show the instability of the former under any XY-type anisotropy at the junction. We analyze the low-temperature spin and the heat conductances. We find evidence of spin fractionalization of the elementary excitations at the four-channel Kondo fixed point by means of the magnetic Wiedemann-Franz law. We caution that the instability under XY anisotropy may hinder the detection of the phenomenology related to the four-channel Kondo effect, therefore requiring careful control in experimental realizations.