  We obtain the complete phase diagram of the hardcore Bose-Hubbard model in
the presence of a period-two superlattice in two and three dimensions. First we
acquire the phase boundaries between the superfluid phase and the `trivial'
insulating phases of the model (the completely-empty and completely-filled
lattices) analytically. Next, the boundary between the superfluid phase and the
half-filled Mott-insulating phase is obtained numerically, using the stochastic
series expansion (SSE) algorithm followed by finite-size scaling. We also
compare our numerical results against the predictions of several approximation
schemes, including two mean-field approaches and a fourth-order strong-coupling
expansion (SCE), where we show that the latter method in particular is
successful in producing an accurate picture of the phase diagram. Finally, we
examine the extent to which several approximation schemes, such as the random
phase approximation and the strong-coupling expansion, give an accurate
description of the momentum distribution of the bosons inside the insulating
phases.
