conservation of bosons.

3.2. Hard-core Boson Model
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The first attempt to interpret a field-induced phase ordering in a BEC was made by Nikuni [@7]. In his work the induced phase diagram of $\rm TlCuCl_{3}$ was studied by applying the Hartree-Fock-Popov (HFP) mean field analysis to the hard-core boson Hamiltonian. For a diluted gas of particles the interatomic interaction is sufficiently weak, therefore the mean field Gross-Pitaevskii theory is a logical tool to study this system. In this model the particle scattering is dominated by the two-body contact interactions, which are described by the $s$-wave scattering length. The physical implication of this condition is that it is highly improbable for more particles to interact with each other simultaneously. In hard-sphere gas, the scattering length is equal to the diameter of the atoms. In the low-density limit only the two particles interactions are important. The low-energy Hamiltonian (3) can be transformed into boson language via identifying $S^{+}=a^{\dagger}$, where $a^{\dagger}$ is the boson creation operator, in $${\cal H} =
  \sum_{{k}}\left(\epsilon-\mu\right))a^{+}_{k}a_{k}+\frac{v_{0}}{V}\sum
  a^{+}_{k+q}a^{+}_{k´-q}a_{k}a_{k´}$$ where $\epsilon = {\hbar k^2}/{2m}$ is the kinetic energy, $v_{0}$ is the short-range repulsion between two magnons that occupy the same site and $V$ is the unit-cell volume of the sample.

