Graphene and the Zermelo optical metric of the BTZ black hole

摘要

It is well known that the low energy electron excitations of the curved graphene sheet σ are solutions of the massless Dirac equation on a 2. +. 1 dimensional ultra-static metric on R×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.