# Interplay between boundary conditions and Wilson's mass in Dirac-like Hamiltonians

## TL;DR

This paper demonstrates how including a minimal $k^2$ Wilson's mass correction in Dirac-like Hamiltonians enables systematic derivation of boundary conditions, improving modeling of finite systems like graphene nanoribbons and topological insulators.

## Contribution

It introduces a method to derive boundary conditions from Wilson's mass corrections, connecting them to Berry-Mondragon boundary conditions for Dirac models.

## Key findings

- Wilson's mass corrections are equivalent to Berry-Mondragon boundary conditions.
- The approach accurately models boundary effects in graphene nanoribbons.
- The method applies to topological crystalline insulators like PbSe.

## Abstract

Dirac-like Hamiltonians, linear in momentum $k$, describe the low-energy physics of a large set of novel materials, including graphene, topological insulators, and Weyl fermions. We show here that the inclusion of a minimal $k^2$ Wilson's mass correction improves the models and allows for systematic derivations of appropriate boundary conditions for the envelope functions on finite systems. Considering only Wilson's masses allowed by symmetry, we show that the $k^2$ corrections are equivalent to Berry-Mondragon's discontinuous boundary conditions. This allows for simple numerical implementations of regularized Dirac models on a lattice, while properly accounting for the desired boundary condition. We apply our results on graphene nanoribbons (zigzag and armchair), and on a PbSe monolayer (topological crystalline insulator). For graphene, we find generalized Brey-Fertig boundary conditions, which correctly describe the small gap seen on \textit{ab initio} data for the metallic armchair nanoribbon. On PbSe, we show how our approach can be used to find spin-orbital-coupled boundary conditions. Overall, our discussions are set on a generic model that can be easily generalized for any Dirac-like Hamiltonian.