  The bilayer Hubbard model describes the antiferromagnet to spin singlet
transition and, potentially, aspects of the physics of unconventional
superconductors. Despite these important applications, significant aspects of
its `phase diagram' in the interplane hopping $t_\perp$ versus on-site
interaction $U$ parameter space, at half filling, are largely in disagreement.
Here we provide an analysis making use of the average sign of weights over the
course of the importance sampling in quantum Monte Carlo simulations to resolve
several central open questions. Specifically, this metric of the weights
clarifies the finite-sized metallic regimes at small $U$. Furthermore, at
strong interactions, it points to the existence of a crossover from a
correlated to uncorrelated band insulator not yet explored in a variety of
existing, unbiased numerical methods. Our work demonstrates the versatility of
using properties of the weights in quantum Monte Carlo simulations to reveal
important physical characteristics of the models under study.
