By implementing the quantum renormalization group method, the effective Hamiltonian of the spin-1 2 Ising model with regularly altering strengths of both the magnetic fields and the exchange couplings are first derived analytically. Then, tripartite quantum discord (QD) and entropic uncertainty (EU) of the Ising model and their quantum phase transitions are investigated. Moreover, the quantum critical points and correlation length exponents of the model are analyzed, and both the block-block EU and the site-site EU are discussed through numerical calculations. It is found that under the system size increasing, the first derivatives of both tripartite QD and block-block EU show nonanalytic and scaling behaviors near the critical point. Furthermore, the strong quantum correlations among multi-particles do not necessarily lead to the decrease of EU, as the site-site EU in the vicinity of the critical point increases due to competitions among quantum correlations. The findings may offer new insights into utilizing EU as an observable of the quantum phase transitions.

By using quantum renormalization group (QRG) approach, we first derive the effective Hamiltonian and QRG equations of the two-dimensional (2D) Ising models with two different time-dependent transverse magnetic fields analytically. Then we examine the nonanalytic and scaling behaviors of the linear-entropy-based uncertainty relation and quantum entanglement of the models near the critical point through numerical analysis. Moreover, we investigate the relation between the quantum critical point and the external magnetic field. Our results show that both the uncertainty relation and the quantum entanglement are feasible to detect the quantum phase transition (QPT), and the uncertainty relation may be a better indicator of QPT than quantum entanglement. Our findings could shed new light on the observable of the QPTs of the solid-state system with the uncertainty relation.