In order to analyse the lattice dependence of ferromagnetism in the two-dimensional Hubbard model we investigate the instability of the fully polarised ferromagnetic ground state (Nagaoka state) on the triangular, honeycomb and kagome lattices. We mainly focus on the local instability, applying single spin flip variational wave functions which include majority spin correlation effects. The question of global instability and phase separation is addressed in the framework of Hartree-Fock theory. We find a strong tendency towards Nagaoka ferromagnetism on the non-bipartite lattices (triangular, kagome) for more than half filling. For the triangular lattice we find the Nagaoka state to be unstable above a critical density of n = 1.887 at ( I = a, thereby significantly improving former variational results. For the kagome lattice the region where ferromagnetism prevails in the phase diagram widely exceeds the flat band regime. Our results wen allow the stability of the Nagaoka state in a small region below half filling. In the case of the bipartite honeycomb lattice several disconnected regions are left for a possible Nagaoka ground state.

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Using a self-consistent mean-eld theory for the S = 1=2 Heisenberg antiferromagnet Kr uger and Schuck recently derived an analytic expression for the dispersion. It is exact in one dimension (d = 1) and agrees well with numerical results in d = 2. With an expansion in powers of the inverse coordination number 1=Z (Z = 2d) we investigate if this expression can be exact for all d. The projection method of Mori-Zwanzig is used for the dynamical spin susceptibility. We nd that the expression of Kr uger and Schuck deviates in order 1=Z 2 from our rigorous result. Our method is generalised to arbitrary spin S and to models with easy-axis anisotropy . It can be systematically improved to higher orders in 1=Z. We clarify its relation to the 1=S expansion.