Motivated by the recent measurement of the activation energy at the quantum Hall state at the filling factor $f=1$ in graphene, we discuss the scaling of the interaction-induced gaps in the vicinity of the Dirac point with the magnetic field. The gap at $f=1$ is shown to be bounded from above by $E\left(1\right)/C$, where $E\left(n\right)=v_F\sqrt{2nB}$ is the Landau-level energy and $C=5.985+O\left(1/N\right)$ is a universal number. The universal scaling functions are computed exactly for a large number of Dirac fermions $N$. We find a sublinear dependence of the gap at the laboratory fields of $10\text{T}<B<50\text{T}$ for realistic values of short-range repulsion between electrons, in quantitative agreement with observation.

• Received 11 April 2008

DOI:https://doi.org/10.1103/PhysRevB.77.245438