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

## TL;DR

This paper develops a theoretical framework to model correlated atomic wires on substrates by mapping them onto effective ladder models, enabling the use of one-dimensional methods to study their electronic properties.

## Contribution

It introduces a novel mapping of 3D substrate-wire systems onto 2D lattices and narrow ladder models, facilitating analysis with established 1D techniques.

## Key findings

- Narrow ladder models accurately represent wires on insulating substrates with at least three legs.
- The approach is validated for noninteracting and Hubbard-interacting electrons.
- Ladders are unsuitable for metallic substrates but effective for insulating ones.

## Abstract

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.