# Full analytical solution of finite-length armchair/zigzag nanoribbons

## TL;DR

This paper provides an exact analytical solution for finite-length armchair and zigzag graphene nanoribbons, including eigenstates, energies, and magnetic properties, with validation against ab initio simulations.

## Contribution

It presents a comprehensive analytical framework for finite graphene nanoribbons, including eigenstates, Coulomb interactions, and magnetic states, validated by ab initio comparisons.

## Key findings

- Exact analytical expressions for eigenstates and energies.
- Identification of magnetic edge states depending on ribbon length.
- Validation of the model through comparison with ab initio simulations.

## Abstract

Finite-length armchair graphene nanoribbons can behave as one dimensional topological materials, that may show edge states in their zigzag-terminated edges, depending on their width and termination. We show here a full solution of Tight-Binding graphene rectangles of any length and width that can be seen as either finite-length armchair or zigzag ribbons. We find exact analytical expressions for both bulk and edge eigen-states and eigen-energies. We write down exact expressions for the Coulomb interactions among edge states and introduce a Hubbard-dimer model to analyse the emergence and features of different magnetic states at the edges, whose existence depends on the ribbon length. We find ample room for experimental testing of our predictions in N = 5 armchair ribbons. We compare the analytical results with ab initio simulations to benchmark the quality of the dimer model and to set its parameters. A further detailed analysis of the ab initio Hamiltonian allows us to identify those variations of the Tight-Binding parameters that affect the topological properties of the ribbons.