matter wave in interacting many-body systems is independent of a detailed microscopic understanding of particles. The intricacies of the many-body problem with interactions that lead to non-separable Hamiltonians are solved by this equation, which introduces effective potentials that are simpler than the original interactions, which in turn renders the physical problem more tractable.

Experimental evidence of the BEC in confined weakly interacting gases was produced by E. A. Cornell, W. Ketterle, and C. E. Wieman in 1995, leading to a Nobel Prize in 2001. That BEC would occur in quantum magnets was first predicted in 1991, and several reports describing real systems were published after 2000 [@7; @13; @14]. The possibility of experimental investigation of the BEC in quantum magnets has led to a deeper understanding of the ground states in strongly correlated systems. Recent studies concerning the BEC of bosons in magnetic systems ranged from a 3$D$ weakly coupled $S$=1/2 dimer compound, $\rm TlCuCl_{3}$ [@7] (the first system in which an identification was attempted with BEC), a quasi-2$D$ spin system, $\rm BaCuSi_{2}O_{6}$ [@13], to a quasi-1$D$ $S$=1 chain with single-ion anisotropy, NiCl$_{2}$-4SC(NH$_{2}$)$_{2}$ (DTN) [@14]. In addition to these three compounds, other gapped spin systems share the same physical picture 