Topological Kondo effect in star junctions of Ising magnetic chains. Exact solution

In this paper I present a conjecture for the Bethe ansatz equations for the model describing a star junction of $M$ quantum critical Ising chains. For $M>3$ such model exhibits the so-called topological Kondo effect [B. Beri and N. R. Cooper, Phys. Rev. Lett. {\bf 109}, 156803 (2012)] related to existence of Majorana zero energy modes at the junction. These modes are of a topological nature; they non-locally encode an SO(M) "spin" which is screened by the collective excitations of the chains. For certain values of $M$ the model is equivalent to the Kondo models with known exact solution. These cases are used to check the validity of the conjecture. It is demonstrated that the model behaves differently for $M$ even and odd; in the former case the model has a Fermi liquid and the latter case corresponds to a non-Fermi liquid infrared fixed point.