  Interactions of collective excitations often lead to rich emergent phenomena
in many-particle quantum systems. In ordered magnets, the elementary
excitations are spin waves (magnons), which obey Bose-Einstein statistics.
Similar to the Cooper pairs in superconductors, magnons can be paired into
bound states under attractive interactions. Even more interestingly, the Zeeman
coupling to a magnetic field acts as a chemical potential that can tune the
particle density through a quantum critical point (QCP), beyond which a
``hidden order'' is predicted to exist. However, experimental confirmation of
this QCP and the associated new state of matter remain elusive. 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 existence of a two-dimensional BEC-QCP at the saturation field.
Inelastic neutron scattering experiments were performed to accurately establish
the magnetic exchange model. An exact solution of the model found stable
2-magnon bound states that were further confirmed by an electron spin resonance
(ESR) experiment, demonstrating that the QCP is due to the pair condensation
and the phase below saturation field is the long-sought-after spin nematic (SN)
phase.

Strongly Correlated Electrons