Abstract
We study many-body ground states for the partial integer fillings of the N=1 Landau level in graphene, by constructing a model that accounts for the lattice scale corrections to the Coulomb interactions. Interestingly, in contrast to the N=0 Landau level, this model contains not only pure delta function interactions but also some of its derivatives. Due to this we find several important differences with respect to the N=0 Landau level. For example at quarter filling when only a single component is filled, there is a degeneracy lifting of the quantum hall ferromagnets and ground states with entangled spin and valley degrees of freedom can become favourable. Moreover at half-filling of the N=1 Landau level, we have found a new phase that is absent in the N=0 Landau level, that combines characteristics of the Kekul\'{e} state and an antiferromagnet. We also find that according to the parameters extracted in a recent experiment, at half-filling of the N=1 Landau level graphene is expected to be in a delicate competition between an AF and a CDW state, but we also discuss why the models for these recent experiments might be missing some important terms.