there is now a field range, from $H_{c1}$ to $H_{c2}$, where the $S^{z}=+1$ level becomes degenerate with the ground state and bosons are created. This gapless phase extends throughout this field range. The coupling to the magnetic field, which is proportional to the magnetization, controls the boson density. The BEC that appears in this intermediate field range corresponds to the coherent superposition of the $S^{z} = +1$ doublet and the $S
= 0$ singlet at each site [@14; @22; @23; @24].

![\[fig4\] Schematic diagram ($T$=0) of the levels of energy. a) The lowest energy levels of $\rm{Ni^{++}}$ in an axial crystalline field as a function of the external magnetic field. b) The broad band indicates dispersion of the lowest excited level due to exchange coupling. Condensation occurs in the range between the fields $H_{c1}$ and $H_{c2}$, in which the magnetization $M$ increases almost linearly up to the saturation.](figure4){width="6cm"}

The transition to the magnetic phase can be adequately described as a condensation of bosons. Three are the basic conditions for BEC induced by a magnetic field: 1) the excited state becoming degenerate with the ground state, 2) strong correlation inducing 3D long-range ordering, and 3) a uniaxial symmetry leading to number 