Probing frustrated spin systems with impurities

We investigate the effective interaction between two localized spin impurities embedded in a frustrated spin-1/2 $J_1\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}{J}_2$ Heisenberg chain. Treating the impurity spins as classical moments coupled locally to the host, we combine second–order perturbation theory with large–scale density matrix renormalization group (DMRG) calculations to determine the impurity–impurity interaction as a function of separation, coupling strength, and magnetic frustration. In the weak–coupling regime, we show that the interaction is governed by the the static spin susceptibility of the host and exhibits oscillatory power–law decay in the gapless phase, modified by universal logarithmic corrections at the SU(2)–symmetric critical point. In the gapped dimerized phase, the interaction decays exponentially with distance. For intermediate and strong impurity–host coupling, we observe a crossover to a boundary–dominated regime characterized by pronounced parity effects associated with the length of the chain segment between impurities, signaling a breakdown of the simple RKKY–like description. Our results establish impurity–impurity interactions as a sensitive probe of frustrated quantum spin liquids and provide a controlled framework for distinguishing gapless and gapped phases through local perturbations.