  Localization of electronic wave functions in modern two-dimensional (2D)
materials such as graphene can impact drastically their transport and magnetic
properties. The recent localization landscape (LL) theory has brought many
tools and theoretical results to understand such localization phenomena in the
continuous setting, but with very few extensions so far to the discrete realm
or to tight-binding Hamiltonians. In this paper, we show how this approach can
be extended to almost all known 2D~lattices, and propose a systematic way of
designing LL even for higher dimension. We demonstrate in detail how this LL
theory works and predicts accurately not only the location, but also the
energies of localized eigenfunctions in the low and high energy regimes for the
honeycomb and hexagonal lattices, making it a highly promising tool for
investigating the role of disorder in these materials.
