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Phase diagram and magnetic collective excitations of the Hubbard model in graphene sheets and layers

We discuss the magnetic phases of the Hubbard model for the honeycomb lattice both in two and three spatial dimensions. A ground state phase diagram is obtained depending on the interaction strength U and electronic density n. We find a first order phase transition between ferromagnetic regions where the spin is maximally polarized (Nagaoka ferromagnetism) and regions with smaller magnetization (weak ferromagnetism). When taking into account the possibility of spiral states, we find that the lowest critical U is obtained for an ordering momentum different from zero. The evolution of the ordering momentum with doping is discussed. The magnetic excitations (spin waves) in the antiferromagnetic insulating phase are calculated from the random-phase-approximation for the spin susceptibility. We also compute the spin fluctuation correction to the mean field magnetization by virtual emission/absorpion of spin waves. In the large $U$ limit, the renormalized magnetization agrees qualitatively with the Holstein-Primakoff theory of the Heisenberg antiferromagnet, although the latter approach produces a larger renormalization.