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Inspired by the recently theoretical development relevant to the experimentaldata of TlCuCl$_3$, particularly those associated with the universal scalingbetween the N\'eel temperature $T_N$ and the staggered magnetization density$M_s$, we carry a detailed investigation of 3-dimensional (3D) dimerizedquantum antiferromagnets using the first principles quantum Monte Carlocalculations. The motivation behind our study is to better understand themicroscopic effects on these scaling relations of $T_N$ and $M_s$, hence toshed some light on some of the observed inconsistency between the theoreticaland the experimental results. Remarkably, for the considered 3D dimerizedmodels, we find that the established universal scaling relations can indeed becategorized by the amount of stronger antiferromagnetic couplings connected toa lattice site. Convincing numerical evidence is provided to support thisconjecture. The relevance of the outcomes presented here to the experiments ofTlCuCl$_3$ is briefly discussed as well.

physical review. b./physical review. b

Classification for the universal scaling of N\'eel temperature and  staggered magnetization density of three-dimensional dimerized spin-1/2  antiferromagnets