The boson number must be set by some external constraint or else bosons will be excitations of the system and vanish as the temperature is lowered to zero, as is the case, e.g., for phonons. In the Hamiltonian (4) every creation operator $a^{\dagger}$ is multiplied by a destruction operator $a$. If the Hamiltonian is rotated by an angle $\phi$ in the plane perpendicular to the field, then $a^{\dagger} \rightarrow
a^{\dagger}e^{i\phi}$ and $a \rightarrow ae^{-i\phi}$, such that $(a^{\dagger}e^{i\phi})(ae^{-i\phi}) = a^{\dagger}a$. Because this Hamiltonian is independent of $\phi$ the number of bosons is conserved. Thus, the uniaxial symmetry of the Hamiltonian creates a number conservation law for the bosons [@29]. In real situations, the square lattice of the crystal can introduce a small anisotropy in the $ab$ plane along with dipole-dipole interactions or Dzyaloshinskii-Moriya (DM) interactions. These effects generally occur at lower energy scales and can therefore be neglected at the temperatures at which DTN was studied. [@21] However, experimental studies should be conducted to discard effects of this possible anisotropy in the formation of BECs.

3.4. Low-temperature Limit and Critical Behavior
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The self-consistent HFP approximation leads to the following effective Hamiltonian for the magnon system: $${\cal H} = \sum_{k}
\left(\epsilon-\mu+{v_{0}}{n}\right)a^{+}_{k}a_{k},$$ 