Efimov Effect in Long-range Quantum Spin Chains

When two non-relativistic particles interact resonantly in three dimensions, an infinite tower of three-body bound states emerges, exhibiting a discrete scale invariance. This universal phenomenon, known as the Efimov effect, has garnered extensive attention across various fields, including atomic, nuclear, condensed matter, and particle physics. In this letter, we demonstrate that the Efimov effect also manifests in long-range quantum spin chains. The long-range coupling modifies the low-energy dispersion of magnons, enabling the emergence of continuous scale invariance for two-magnon states at resonance. This invariance is subsequently broken to discrete scale invariance upon imposing short-range boundary conditions for the three-magnon problem, leading to the celebrated Efimov bound states. Using effective field theory, we theoretically determine how the ratio of two successive binding energies depends on the interaction range, which agrees with the numerical solution of the bound-state problem. We further discuss generalizations to arbitrary spatial dimensions, where the traditional Efimov effect serves as a special case. Our results reveal universal physics in dilute quantum gases of magnons that can be experimentally tested in trapped-ion systems.