This essay summarizes the well known work of Gonzalez, Guinea, Vozmediano, Novoselov, Geim and others according to which the effective long-distance theory of the half-filled Hubbard model on the honeycomb lattice is given by the Dirac equation for a massless quasiparticle, along with a gauge field which can be used to model valley degrees of freedom and the effect of defects and disorder in the lattice on the quasi-particle dynamics. This fact was known as far back as 1992, see arXiv:cond-mat/9208004.
There a few other facts not mentioned in the essay but which are important for this topic:
1. In arXiv:0909.3057, Gonzalez and Herrero have also shown how one can model a Dirac particle in the presence of a wormhole background in a graphene based setup.
2. It is also well known (see for e.g. arXiv:gr-qc/9405070) that gravity in 2+1 dimensions is a purely topological theory with an equivalent description as a theory of a Chern-Simons gauge field. Consequently a quantum hall system - in which the CS theory plays an integral part - can be utilized as a substrate for tests of quantum gravity.
3. The entropy of a black hole horizon also is given by a Chern-Simons theory in the LQG approach (e.g. arXiv:gr-qc/9710007). Thus a quantum hall system could plausibly be used to model an isolated horizon.
4. Kitaev has also done a great deal of work (arXiv:cond-mat/0506438) in showing how non-abelian anyons in a hexagonal lattice can be used for quantum computation.
In this way, three different threads of theoretical physics - Chern-Simons theory describing the quantum hall effect, 2+1 dimensional quantum gravity and quantum computation on a lattice come together quite naturally in an experimentally accessible setup as simple as that of graphene.
The essay itself is a very readable summary of such efforts and a pedagogical explanation of this line of research. In fact, this topic can use all the publicity it can get and I am happy to see that happen. Congratulations on this honor, Tobias!