Emergent Horava gravity in graphene

First of all, we reconsider the tight-binding model of monolayer graphene, in which the variations of the hopping parameters are allowed. We demonstrate that the emergent 2D$2D$ Weitzenbock geometry as well as the emergent U(1)$U\left(1\right)$ gauge field appear. The emergent gauge field is equal to the linear combination of the components of the zweibein. Therefore, we actually deal with the gauge fixed version of the emergent 2+1$2+1$   D$D$ teleparallel gravity. In particular, we work out the case, when the variations of the hopping parameters are due to the elastic deformations, and relate the elastic deformations with the emergent zweibein. Next, we investigate the tight-binding model with the varying intralayer hopping parameters for the multilayer graphene with the ABC$ABC$ stacking. In this case the emergent 2D$2D$ Weitzenbock geometry and the emergent U(1)$U\left(1\right)$ gauge field appear as well, and the emergent low energy effective field theory has the anisotropic scaling.