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.