  For arbitrary space dimension $d$ we investigate the quantum phase
transitions of two paradigmatic spin models defined on a hypercubic lattice,
the coupled-dimer Heisenberg model and the transverse-field Ising model. To
this end high-order linked-cluster expansions for the ground-state energy and
the one-particle gap are performed up to order 9 about the decoupled-dimer and
high-field limits, respectively. Extrapolations of the high-order series yield
the location of the quantum phase transition and the correlation-length
exponent $\nu$ as a function of space dimension $d$. The results are
complemented by $1/d$ expansions to next-to-leading order of observables across
the phase diagrams. Remarkably, our analysis of the extrapolated linked-cluster
expansion allows to extract the coefficients of the full $1/d$ expansion for
the phase-boundary location in both models exactly in leading order and
quantitatively for subleading corrections.

Strongly Correlated Electrons