The Hierarchical Graphene model

Abstract

The hierarchical graphene model is a simple toy model which is useful to understand the mechanics of renormalization group flows in super-renormalizable systems. It is based on a model of interacting electrons in graphene, for which the renormalization group analysis was carried out by Giuliani and Mastropietro. The analysis of the hierarchical graphene model is significantly simpler than graphene, but one should not expect it to produce good quantitative results about real-world graphene. Rather, the hierarchical model is useful as a teaching tool to understand the core concepts of renormalization group techniques. In this paper, we will first introduce a model for electrons in graphene and set it up for a renormalization group treatment by introducing its Grassmann representation and scale decomposition. We then define the hierarchical graphene model and study it's renormalization group flow. From a renormalization group point of view, graphene is quite simple: it is super-renormalizable. As an illustration of a more complicated system, we repeat the analysis for the Kondo model, which is a strongly coupled model with a non-trivial fixed point.