  An interacting spinless fermion wire coupled to a three-dimensional (3D)
semiconducting substrate is approximated by a narrow ladder model (NLM) with
varying number of legs. We compute density distributions, gaps,
charge-density-wave (CDW) order parameters, correlation functions, and the
central charge using the density-matrix renormalization group method. Three
ground-state phases are observed: a one-component Luttinger liquid, a
quasi-one-dimensional (1D) CDW insulator, and a band insulator. We investigated
the convergence of the NLM properties with increasing number of legs
systematically and confirm that the NLM is a good approximation for the
quasi-1D phases (Luttinger liquid and CDW) of the 3D wire-substrate model. The
quantum phase transitions between these phases are investigated as function of
the coupling between wire and substrate. The critical nearest-neighbor
interaction increases with increasing coupling between wire and substrate and
thus the substrate stabilizes the Luttinger liquid in the wire. Our study
confirms that a Luttinger liquid or CDW insulator phase could occur in the
low-energy properties of atomic wires deposited on semiconducting substrates.
