Topological low-energy modes in image Landau levels of graphene: A possibility of a quantum-liquid ground state

We point out that the zero-energy Landau level of Dirac fermions in graphene can be, in the presence of a repulsive electron–electron interaction, split into two (levels) associated with a “bond ordering” formation having a “Kekulé pattern”, which respects the chiral symmetry. Since the Kekulé pattern has a three-fold degeneracy, domain structures are implied, for which we show that in-gap states localized along the domain boundaries exist as topological states. Based on this a possibility of a quantum-liquid ground state of graphene in magnetic fields is discussed.