Many-Body Orbital Paramagnetism in Doped Graphene Sheets

The orbital magnetic susceptibility of a gas of noninteracting massless Dirac fermions is zero when the Fermi energy is away from the Dirac point. Making use of diagrammatic perturbation theory, we calculate exactly the orbital magnetic susceptibility of massless Dirac fermions to first order in the Coulomb interaction demonstrating that it is finite and positive. Doped graphene sheets are thus unique systems in which the orbital magnetic susceptibility is completely controlled by many-body effects.

Figure 1

Diagrams for the orbital magnetic susceptibility. (a) The noninteracting bubble. (b) The first-order vertex correction. (c) and (d) The two first-order self-energy diagrams. The wave-vector labeling of the propagators ensures time-reversal symmetry and thus guarantees that the two self-energy diagrams give an identical contribution.

Figure 2

The dimensionless quantity $\mathcal{N}=-({\Xi }_1+{\Xi }_2+{\Xi }_3)/(2\pi )$ as a function of the fine-structure constant ${\alpha }_{\mathrm{ee}}$. Note that $\mathcal{N}({\alpha }_{\mathrm{ee}})>0$ and that it depends weakly on ${\alpha }_{\mathrm{ee}}$. Inset: a zoom for very small ${\alpha }_{\mathrm{ee}}$. The horizontal axis is in logarithmic scale. The solid line represents the analytical weak-coupling result, $\mathcal{N}({\alpha }_{\mathrm{ee}}\to 0)\to -\ln ({\alpha }_{\mathrm{ee}})/(48{\pi }^2)$.