Quantum Hall effect in graphene: Emergent modular symmetry and the semicircle law

Low-energy transport measurements in quantum Hall systems have been argued to be governed by emergent modular symmetries whose predictions are robust against many of the detailed microscopic dynamics. We propose the recently observed quantum Hall effect in graphene as a test of these ideas, and identify to this end a class of predictions for graphene which would follow from the same modular arguments. We are led to a suite of predictions for high-mobility samples that differs from those obtained for the conventional quantum Hall effect in semiconductors, including predictions for the locations of the quantum Hall plateaus, predictions for the positions of critical points on transitions between plateaus, a selection rule for which plateaus can be connected by low-temperature transitions and a semicircle law for conductivities traversed during these transitions. Many of these predictions appear to provide a good description of graphene measurements performed with intermediate-strength magnetic fields.