We investigate the ground state properties of the two dimensional half-filled one band Hubbard model in the strong (large-U ) to intermediate coupling limit (i.e. away from the strict Heisenberg limit) using an effective spin-only low-energy theory that includes nearest-neighbor exchange, ring exchange, and all other spin interactions to order t(t/U ) 3 . We show that the operator for the staggered magnetization, transformed for use in the effective theory, differs from that for the order parameter of the spin model by a renormalization factor accounting for the increased charge fluctuations as t/U is increased from the t/U → 0 Heisenberg limit. These charge fluctuations lead to an increase of the quantum fluctuations over and above those for an S = 1/2 antiferromagnet. The renormalization factor ensures that the zero temperature staggered moment for the Hubbard model is a monotonously decreasing function of t/U , despite the fact that the moment of the spin Hamiltonien, which depends on transverse spin fluctuations only, in an increasing function of t/U . We also comment on quantitative aspects of the t/U and 1/S expansions.

Spin-only descriptions of the half-filled one-band Hubbard model are relevant for a wide range of Mott insulators. In addition to the usual Heisenberg exchange, many other types of interactions, including ring exchange, appear in the effective Hamiltonian in the intermediate coupling regime. In order to improve on the quantitative description of magnetic excitations in the insulating antiferromagnetic phase of copper-oxide ͑cuprate͒ materials, and to be consistent with band-structure calculations and photoemission experiments on these systems, we include second-and third-neighbor hopping parameters, tЈ and tЉ, into the Hubbard Hamiltonian. A unitary transformation method is used to find systematically the effective Hamiltonian and any operator in the spin-only representation. The results include all closed four-hop electronic pathways in the canonical transformation. The method generates many ring exchange terms that play an important role in the comparison with experiments on La 2 CuO 4 . Performing a spin-wave analysis, we calculate the magnon dispersion as a function of U, t, tЈ, and tЉ. The four parameters are estimated by fitting the magnon dispersion to the experimental results of Coldea et al. ͓Phys. Rev. Lett. 86, 5377 ͑2001͔͒ for La 2 CuO 4 . The ring exchange terms are found essential, in particular to determine the relative sign of tЈ and tЉ, with the values found in good agreement with independent theoretical and experimental estimates for other members of the cuprate family. The zero-temperature sublattice magnetization is calculated using these parameters and also found to be in good agreement with the experimental value estimated by Lee et al. ͓Phys. Rev. B 60, 3643 ͑1999͔͒. We find a value of the interaction strength U Ӎ 8t consistent with Mott insulating behavior.