Probing quantum critical crossover via impurity renormalization group

Quantum impurities can host exotic many-body states that serve as sensitive probes of bath correlations. However, quantitative and non-perturbative methods for determining impurity thermodynamics in such settings remain scarce. Here, we introduce an impurity renormalization group approach that merges the tensor-network representation with the numerical renormalization group cutoff scheme. This method overcomes conventional limitations by treating bath correlations and impurity interactions on an equal footing. Applying our approach to the finite-temperature quantum critical regime of quantum spin systems, we uncover striking impurity-induced phenomena. In a coupled Heisenberg ladder, the impurity triggers a fractionalization of the local magnetic moment. Moreover, the derivative of the impurity susceptibility develops cusps that mark the crossover into the quantum critical regime. We also observe an exotic evolution of the spin correlation function driven by the interplay between bath correlations and the impurity. Our results demonstrate that this method can efficiently solve correlated systems with defects, opening new pathways to discovering novel impurity physics beyond those in non-interacting thermal baths.