We theoretically study electronic properties of a graphene sheet on xy plane in a spatially nonuniform magnetic field, $B = B_0 hat{z}$ in one domain and $B = B_1 hat{z}$ in the other domain, in the quantum Hall regime and in the low-energy limit. We find that the magnetic edge states of the Dirac fermions, formed along the boundary between the two domains, have features strongly dependent on whether $B_0$ is parallel or antiparallel to $B_1$. In the parallel case, when the Zeeman spin splitting can be ignored, the magnetic edge states originating from the $n=0$ Landau levels of the two domains have dispersionless energy levels, contrary to those from the $n e 0$ levels. Here, $n$ is the graphene Landau-level index. They become dispersive as the Zeeman splitting becomes finite or as an electrostatic step potential is additionally applied. In the antiparallel case, the $n=0$ magnetic edge states split into electron-like and hole-like current-carrying states. The energy gap between the electron-like and hole-like states can be created by the Zeeman splitting or by the step potential. These features are attributed to the fact that the pseudo-spin of the magnetic edge states couples to the direction of the magnetic field. We propose an Aharonov-Bohm interferometry setup in a graphene ribbon for experimental study of the magnetic edge states.
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