In this hard-core model applied to the Ni$^{2+}$ ions the density of the magnetization along the field direction, $S^{z}$ = +1, is mapped onto the boson density. Here the $S^z=-1$ component is neglected, because we are interested in low-field and low-temperatures regions, where this level does not contribute. An analysis taking this level into account is discussed in Section 4.17 [@14; @22].

The physics of the magnons depends on the relative strengths of the repulsive interaction $v_{0}$ and kinetic energy. When the interaction is attractive, $v_{0} < 0$, the system collapses. If the repulsion interaction dominates, the system evolves into a state where the bosons form a large lattice that gives rise to a magnetization plateau in the $M$ $vs$. $H$ curve. If the kinetic term dominates, the system, which corresponds to a Mott insulator in the low-field region, undergoes a condensation at the first critical field $H_{c1}$. Above this field the density of bosons increases proportionally to the longitudinal component of the magnetization $M$ up to the saturation value, $M_{sat}$, at the second critical field $H_{c2}$. The result is an antiferromagnetic state characterized by the presence of a staggered transverse moment, $M_{sta}$, perpendicular to the applied magnetic field, forming 