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As a prototype model of antiferromagnetism, we propose a repulsive Hubbard Hamiltonian defined on a\udgraph\udL\ud5\udA\ud̄\udB\udwith\udA\ud̆\udB\ud5\udB\ud, and bonds connecting any element of\udA\udwith all the elements of\udB\ud. Since all\udthe hopping matrix elements associated with each bond are equal, the model is invariant under an arbitrary\udpermutation of the\udA\udsites and/or of the\udB\udsites. This is the Hubbard model defined on the so-called (\udN\udA\ud,\udN\udB\ud)\udcomplete bipartite graph,\udN\udA\ud(\udN\udB\ud) being the number of elements in\udA\ud(\udB\ud). In this paper, we analytically find\udthe\udexact\udground state for\udN\udA\ud5\udN\udB\ud5\udN\udat half filling for any\udN\ud; the repulsion has a maximum at a critical\udN\ud-dependent value of the on-site Hubbard\udU\ud. The wave function and the energy of the unique, singlet ground\udstate assume a particularly elegant form for\udN\ud!\ud`\ud. We also calculate the spin-spin correlation function and\udshow that the ground state exhibits an antiferromagnetic order for any nonzero\udU\udeven in the thermodynamic\udlimit. We are aware of no previous explicit analytic example of an antiferromagnetic ground state in a\udHubbard-like model of itinerant electrons. The kinetic term induces nontrivial correlations among the particles,\udand an antiparallel spin configuration in the two sublattices becomes energetically favored at zero temperature.\udOn the other hand, if the thermodynamic limit is taken and then zero temperature is approached, a paramag-\udnetic behavior results. The thermodynamic limit does not commute with the zero-temperature limit, and this\udfact can be made explicit by the analytic solutions

Antiferromagnetism in the exact ground state of the half-filled Hubbard model on the complete bipartite graph