  We study a system of hard-core bosons at half-filling in a one-dimensional
optical superlattice. The bosons are allowed to hop to nearest and next-nearest
neighbor sites producing a zig-zag geometry and we obtain the ground state
phase diagram as a function of microscopic parameters using the finite-size
density matrix renormalization group (FS-DMRG) method. Depending on the sign of
the next-nearest neighbor hopping and the strength of the superlattice
potential the system exhibits three different phases, namely the bond-order
(BO) solid, the superlattice induced Mott insulator (SLMI) and the superfluid
(SF) phase. When the signs of both hopping amplitudes are the same (the
"unfrustrated" case), the system undergoes a transition from the SF to the SLMI
at a non-zero value of the superlattice potential. On the other hand, when the
two amplitudes differ in sign (the "frustrated" case), the SF is unstable to
switching on a superlattice potential and also exists only up to a finite value
of the next nearest neighbor hopping. This part of the phase diagram is
dominated by the BO phase which breaks translation symmetry spontaneously even
in the absence of the superlattice potential and can thus be characterized by a
bond order parameter. The transition from BO to SLMI appears to be first order.
