We employ the stochastic series expansion quantum Monte Carlo method to investigate the
S=1/2 antiferromagnetic Heisenberg model on a bilayer square lattice with diluted interlayer couplings. Both regular and random dilution patterns are considered. In systems with regular dilution, tuning the interlayer interaction drives a quantum phase transition from a N\'eel-ordered phase to a quantum disordered phase, consistent with the
O(3) universality class. In contrast, random dilution gives rise to a two-step transition: from the N\'eel phase to an intermediate Mott glass (MG) phase, followed by a transition to the quantum disordered phase. Within the MG phase, the uniform magnetic susceptibility exhibits a stretched-exponential temperature dependence ${\ensuremath{\chi}}_{u}\ensuremath{\sim}exp(\ensuremath{-}b/{T}^{\ensuremath{\alpha}}), 0<\ensuremath{\alpha}<1$. At the N\'eel-to-glass transition, the quenched disorder modifies the critical exponents in a manner consistent with the Harris criterion. These findings provide new insights into disorder-driven quantum phase transitions and the emergence of glassy phases in diluted bilayer quantum magnets.
We review the theoretical behaviour of the total and one-particle structure factors at a quantum phase transition for temperature T = 0. The predictions are compared with exact or numerical results for the transverse Ising model, the alternating Heisenberg chain, and the bilayer Heisenberg model. At the critical wavevector, the results are generally in accord with theoretical expectations. Away from the critical wavevector, however, different models display quite different behaviours for the one-particle residues and structure factors.