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

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