Correlated atomic wires on substrates. I. Mapping to quasi-one-dimensional models

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.