a BEC state. The absolute value and relative angle of $M_{sta}$ are related to the amplitude and the phase of the condensate wave function, respectively. In this phase the condensate shows a macroscopic order parameter that spontaneously breaks global phase symmetry by the emerging antiferromagnetic longe-range order. In these conditions, there must exist a gapless mode called Nambu-Goldstone in the new phase, which is necessary to keep the original invariance of the Hamiltonian. At $H_{c2}$ all sites are occupied by bosons and the system enters a second Mott insulating phase [@25; @26; @27; @28].

3.3. Boson Number Conservation
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At the induced phase the Ni magnetic spins do not have any preferred orientation in the plane perpendicular to the magnetic field, and the antiferromagnetic order is XY-like. The tetragonal crystal structure provides a uniaxial crystal-field symmetry about the direction of the applied field. It is this symmetry that enforces number conservation among the bosons. This forces the bosons to remain in the system and macroscopically occupy the ground state at low temperatures. Boson-number conservation is a key condition that separates bosonic systems that condense from those that do not. Formally, the phenomenon of BEC requires the conservation of particle number. 