Structure and the Lamb-shift-like quantum splitting of the pseudo-zero-mode Landau levels in bilayer graphene

In a magnetic field, bilayer graphene supports an octet of zero-energy Landau levels with an extra twofold degeneracy in Landau orbitals $n=0$ and 1. It is shown that this orbital degeneracy is lifted due to Coulombic quantum fluctuations of the valence band (the Dirac sea); this is a quantum effect analogous to the Lamb shift in the hydrogen atom. A detailed study is made of how these zero-energy levels evolve, with filling, into a variety of pseudo-zero-mode Landau levels in the presence of possible spin and valley breaking and Coulomb interactions, and a comparison is made with experimental results.