Graphene and the Zermelo optical metric of the BTZ black hole

It is well known that the low energy electron excitations of the curved graphene sheet ΣΣ are solutions of the massless Dirac equation on a 2+12+1 dimensional ultra-static metric on R×ΣR×Σ. An externally applied electric field on the graphene sheet induces a gauge potential which could be mimicked by considering a stationary optical metric of the Zermelo form, which is conformal to the BTZ black hole when the sheet has a constant negative curvature. The Randers form of the metric can model a magnetic field, which is related by a boost to an electric one in the Zermelo frame. We also show that there is a fundamental geometric obstacle to obtaining a model that extends all the way to the black hole horizon.

► Electrons on graphene sheets obey the Dirac equation in ultra-static metrics. ► External magnetic fields on give ultra-stationary metrics. ► These metrics are of optical-Zermelo form. ► Embedded surfaces corresponding to BTZ black holes are given.