the number of particles is conserved, their characteristics almost prevent these systems from becoming BECs. All these systems are believed to be defective condensates.

Recently it has been reported that a gas of magnons at room temperature can be continuously overpopulating the lowest energy level, which shows characteristics of BEC, when it is driven by microwave pumping.[@18; @19]

On the other hand, diluted magnons in equilibrium can be pure condensates at low temperatures, just like cold gas.

2.2. BEC of Magnons
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Magnetic insulators with magnetic ions are characterized by a gap separating the singlet ground state from the lowest-energy excited state. If the gap is closed by the Zeeman effect, the resulting quasiparticles can undergo the Bose-Einstein condensation transition.

The BEC phase in quantum magnets is an XY antiferromagnetic state in which the spins spontaneously choose a particular orientation in the XY plane. The number of bosons is proportional to the net magnetization, i. e., the $S^{z}$ component of the spins. On the other hand, the number of condensed bosons is proportional to the antiferromagnetically ordered $S^{x}$ component of the spins. Thus the XY antiferromagnetic order parameter maps directly onto the order parameter for Bose-Einstein condensation. It has a 