Intrinsic Zeeman Effect in Graphene

The intrinsic Zeeman energy is precisely one half of the cyclotron energy for electrons in graphene. As a result a Landau-level mixing occurs to create the energy spectrum comprised of the $4j$-fold degenerated zero-energy level and 4-fold degenerated nonzero-energy levels in the $j$-layer graphene, where $j=1,2,3$ for monolayer, bilayer and trilayer, respectively. The degeneracy manifests itself in the quantum Hall (QH) effect. We study how the degeneracy is removed by the Coulomb interactions. With respect to the zero-energy level, an excitonic gap opens by making a BCS-type condensation of electron-hole pairs at the filling factor $\nu =0$. It gives birth to the Ising QH ferromagnet at $\nu =\pm 1$ for monolayer, $\nu =\pm 1,\pm 3$ for bilayer, and $\nu =\pm 1,\pm 3,\pm 5$ for trilayer graphene from the zero-energy degeneracy. With respect to the nonzero-energy level, a remarkable consequence is derived that the effective Coulomb potential depends on spins, since a single energy level contains up-spin and down-spin electrons belonging to different Landau levels. The spin-dependent Coulomb interaction leads to the valley polarization at $\nu =\pm 4, \pm 8, \pm 12, ...$ for monolayer, $\nu =\pm 2, \pm 6, \pm 10, >...$ for bilayer, and $\nu =\pm 2,\pm 4, \pm 8, \pm 12, ...$ for trilayer graphene.