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Hierarchy of Spin and Valley Symmetry Breaking in Quantum Hall Single Layer Graphene

We explore several microscopic mechanisms for breaking the $n=0$ fourfold Landau level degeneracy in a single-layer graphene. Valley-scattering random potential, Zeeman interaction, and electron-phonon coupling are considered in the presence of SU(4)-symmetric Coulomb exchange interaction. Among all the mechanisms considered, it is the electron-phonon coupling combined with the Zeeman interaction which leads to the full splitting of the $n=0$ Landau levels. A recent controversy of "valley-first" or "spin-first" breaking of SU(4) symmetry of the $n=0$ graphene Landau level is examined in light of our results. Existence of midgap states between Landau levels of opposite valley polarity are demonstrated.