Dynamics and thermodynamics of quantum impurities at the edge of the spin chains

We investigate the overscreening of a spin-1/2 impurity in the Heisenberg-Babujian spin-s chain - an integrable generalization of the conventional Heisenberg spin-1/2 chain. Depending on the ratio between the bulk and boundary coupling strengths, the model exhibits distinct boundary phases characterized by different thermodynamic and dynamical behaviors. Using Bethe ansatz and tensor-network techniques, we analyze the thermodynamic properties and demonstrate that the Hilbert space falls into "towers of states," corresponding to bulk excitations built upon different boundary configurations. This structure gives rise to a non-monotonic dependence of the impurity entropy. It also manifests itself in the dynamic properties of the system as evidenced by the tensor-network calculation of the impurity spectral function, which reveals abrupt qualitative changes across the boundary phase transitions.