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We develop a large-N expansion for Gutzwiller projected spin states. We consider valence-bond singlets, constructed by Schwinger bosons or fermions, wich are variational ground states for quantum antiferromagnets. This expansion is simpler than the familiar expansions of the quantum Heisenberg model, and thus more instructive. The diagrammatic rules of this expansion allow us to prove certain identities to all orders in 1/N. We derive the on-site spin fluctuations sum rule for arbitrary N. We calculate the correlations of the one-dimensional valence-bond solid states and the Gutzwiller projected Fermi gas up to order 1/N. For the boson case, we are surprised to find that the mean-field, the order-1/N, and the exact correlations are simply proportional. For the fermion case, the 1/N correction enhances the zone-edge singularity. The comparison of our leading-order terms to known results for N-2 enhances our iunderstanding of large-N approximations in general.
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