Bose-Einstein condensation of a two-magnon bound state in a spin-one triangular lattice

In ordered magnets, the elementary excitations are spin waves (magnons), which obey Bose-Einstein statistics. Similarly to Cooper pairs in superconductors, magnons can be paired into bound states under attractive interactions. The Zeeman coupling to a magnetic field is able to tune the particle density through a quantum critical point (QCP), beyond which a "hidden order" is predicted to exist. Here we report direct observation of the Bose-Einstein condensation (BEC) of the two-magnon bound state in Na$_2$BaNi(PO$_4$)$_2$. Comprehensive thermodynamic measurements confirmed the two-dimensional BEC-QCP at the saturation field. Inelastic neutron scattering experiments were performed to establish the microscopic model. An exact solution revealed stable 2-magnon bound states that were further confirmed by electron spin resonance and nuclear magnetic resonance experiments, demonstrating that the QCP is due to the pair condensation and the phase below saturation field is likely the long-sought-after spin nematic phase.