Phase diagram of an impurity in the spin-1/2 chain: two channel Kondo effect versus Curie law

We consider a magnetic s=1/2 impurity in the antiferromagnetic spin chain as a function of two coupling parameters: the symmetric coupling of the impurity to two sites in the chain $J_1$ and the coupling between the two sites $J_2$. By using field theory arguments and numerical calculations we can identify all possible fixed points and classify the renormalization flow between them, which leads to a non-trivial phase diagram. Depending on the detailed choice of the two (frustrating) coupling strengths, the stable phases correspond either to a decoupled spin with Curie law behavior or to a non-Fermi liquid fixed point with a logarithmically diverging impurity susceptibility as in the two channel Kondo effect. Our results resolve a controversy about the renormalization flow.