Universal bound states in long-range spin chains with an impurity

Understanding how quasi-particles interact with impurities is crucial for unveiling novel properties of quantum many-body systems. A prominent example is the enhanced scattering between electrons and magnetic impurities in the low-energy limit, which gives rise to the Kondo effect. In this letter, motivated by recent developments in quantum simulation platforms, we investigate the universal behavior of long-range quantum spin chains with a single local impurity, focusing on systems that conserve the magnon number. Using effective field theory, we show that distinct classes of universal three-magnon states can emerge when the impurity-mediated two-magnon interaction is on resonance. When the long-range coupling decays as $1/r^α$, we find that (i) for $α\in (2,2.89)$, the system exhibits Efimov effects, with the three-body binding energy forming a geometric series $\ln|E^{(n)}_{\text{3-body}}|\sim-n (α-1) π/s_0(α)$, (ii) for $α=2$, the system shows semi-super Efimov effects with $\ln|E^{(n)}_{\text{3-body}}|\sim-(nπ-θ)^2/8$. Our theoretical prediction is validated by the numerical solution of the Skorniakov-Ter-Martirosian equation. Our results could be tested experimentally in the future on quantum simulation platforms.