  We present a theoretical study of two spinless fermion wires coupled to a
three dimensional semiconducting substrate. We develop a mapping of wires and
substrate onto a system of two coupled two-dimensional ladder lattices using a
block Lanczos algorithm. We then approximate the resulting system by narrow
ladder models, which can be investigated using the density-matrix
renormalization group method. In the absence of any direct wire-wire hopping we
find that the substrate can mediate an effective wire-wire coupling so that the
wires could form an effective two-leg ladder with a Mott charge-density-wave
insulating ground state for arbitrarily small nearest-neighbor repulsion. In
other cases the wires remain effectively uncoupled even for strong
wire-substrate hybridizations leading to the possible stabilization of the
Luttinger liquid phase at finite nearest-neighbor repulsion as found previously
for single wires on substrates. These investigations show that it may be
difficult to determine under which conditions the physics of correlated
one-dimensional electrons can be realized in arrays of atomic wires on
semiconducting substrates because they seem to depend on the model (and
consequently material) particulars.
