  We present a theoretical study of correlated atomic wires deposited on
substrates in two parts. In this first part, we propose lattice models for a
one-dimensional quantum wire on a three-dimensional substrate and map them onto
effective two-dimensional lattices using the Lanczos algorithm. We then discuss
the approximation of these two-dimensional lattices by narrow ladder models
that can be investigated with well-established methods for one-dimensional
correlated quantum systems, such as the density-matrix renormalization group or
bosonization. The validity of this approach is studied first for noninteracting
electrons and then for a correlated wire with a Hubbard electron-electron
repulsion using quantum Monte Carlo simulations. While narrow ladders cannot be
used to represent wires on metallic substrates, they capture the physics of
wires on insulating substrates if at least three legs are used. In the second
part [arXiv:1704.07359], we use this approach for a detailed numerical
investigation of a wire with a Hubbard-type interaction on an insulating
substrate.
