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Particle-hole symmetry and bifurcating ground state manifold in the quantum Hall ferromagnetic states of multilayer graphene

The orbital structure of the quantum Hall ferromagnetic states in the zero-energy Landau level in chiral multilayer graphene (AB, ABC, ABCA, etc.\ stackings) is determined by the exchange interaction with all levels, including deep-lying states in the Dirac sea. This exchange field favors orbitally coherent states with a U(1) orbital symmetry if the filling factor $ν$ is not a multiple of the number of layers. If electrons fill the orbital sector of a fixed spin/valley component to one-half, e.g., at $ν=\pm3,\pm1$ in the bilayer and at $ν=\pm2,\pm6$ in the ABCA four-layer, there is a transition to an $Z_2\times$U(1) manifold. For weak interaction, the structure in the zero-energy Landau band compensates for the different exchange interaction on the sublattices in the Landau orbitals; on the other side, the ground state comes in two copies that distribute charge on the sublattices differently. We expect a sequence of similar bifurcations in multilayers of Bernal stacking.