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 $1/r^\alpha$, we find that (i) for $\alpha\in (2,2.89)$, the system exhibits Efimov effects, with the three-body binding energy forming a geometric series $\ln|E^{(n)}_{\text{3-body}}|\sim-n (\alpha-1) \pi/s_0(\alpha)$, (ii) for $\alpha=2$, the system shows semi-super Efimov effects with $\ln|E^{(n)}_{\text{3-body}}|\sim-(n\pi-\theta)^2/8$. 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.