Parallel transport of electrons in graphene parallels gravity

Abstract

Geometrically a crystal containing dislocations and disclinations can be envisaged as a ``fixed frame'' Cartan-Einstein space-time carrying torsion and curvature, respectively. We demonstrate that electrons in defected graphene are transported in the same way as fundamental Dirac fermions in a nontrivial $2+1$-dimensional space-time, with the proviso that the graphene electrons remember the lattice constant through the valley quantum numbers. The extra ``valley holonomy'' corresponds to modified Euclidean symmetry generators.