Scattering quantum walk framework for two-dimensional materials: The case of honeycomb lattice structures

Bidimensional crystals display unique properties of both fundamental and applied interest, with a good part of these properties being related to the topological aspects of 2D materials. Discrete quantum walks models, commonly used in the area of quantum information, are mathematical constructions in which the underlying network topology plays a fundamental role in determining the systems behavior. Here we present a complete scattering quantum walks approach to study 2D honeycomb lattice problems, the structure of paradigmatic 2D Dirac materials like graphene, germanene and silicene. The framework great flexibility relies on considering two arbitrary $3\ifmmode\times\else\texttimes\fi{}3$ unitary scattering matrices ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}}^{(\ifmmode\pm\else\textpm\fi{})}$ to describe the local dynamics in the lattice fundamental cell. From a simple analytic choice for ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}}^{(\ifmmode\pm\else\textpm\fi{})}$, we address important aspects of 2D materials like transport characteristics. We also readily obtain analytic formulas for quantities which are commonly derived from a tight-binding approximation. Most importantly, we derive a rather general equation for the system energy bands based on the determinant of products of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}}^{(\ifmmode\pm\else\textpm\fi{})}$. We show that by properly setting these matrices (numerically), we get good agreements between our calculations for the $\ensuremath{\pi}$ and ${\ensuremath{\pi}}^*$ energy bands of the graphene, germanene and siliciene with accurate ab initio methods in the literature. We finally briefly discuss how the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}}^{(\ifmmode\pm\else\textpm\fi{})}$ could be computed from first principles, making the present an useful protocol to investigate 2D materials.