Geometrical and topological aspects of graphene and related materials

Graphene, a two-dimensional crystal made of carbon atoms, provides a new and unexpected bridge between low- and high-energy physics. The field has evolved very quickly and there are already a number of good reviews available in the literature. Graphene constitutes a condensed-matter realization of lower dimensional quantum field theory models that were proposed to confront important-still unresolved-puzzles in the area: chiral symmetry breaking and quark confinement. The new materials named topological insulators, closely related to graphene, are physical realizations of topological field theory. This article reviews some of these topics with the aim of bridging the gap and making these condensed-matter issues accessible to high-energy readers. The electronic interactions in the monolayer are analyzed with special emphasis on the recent experimental confirmation of some theoretical predictions. The issue of spontaneous chiral symmetry breaking in the model materials is also reviewed. Finally we give an extensive description of some recent topological properties of graphene that allow us to understand the main aspects of topological insulators.