Exact mean-field Hamiltonian for fermionic lattice models in high dimensions

We show that even for fermionic lattice models with on-site interaction a mean-field Hamiltonian can be constructed, which--in analogy with spin-lattice models--becomes exact in the limit of high spatial dimensions d. Here the mean fields are fermion operators, rather than numbers. We use this method to obtain the exact solution of a simplified Hubbard model on a Bethe lattice for all temperatures. The order parameter is found to have the conventional mean-field form, but exhibits unusually rich behavior as a function of the interaction.