Exact Quadratic Effective Hamiltonian for the Hubbard Model and its Exact Solutions

Applying the recently developed self-consistent effective Hamiltonian theory (SCEHT) to the Hubbard model, we derive the exact quadratic effective Hamiltonian for the Hubbard model. The exact solutions of the effective Hamiltonian is presented. We showed that for the Hubbard model, when no-double occupancy constraint on the b-fermions are satisfied exactly, the corresponding eigenstates, the vacuum state in particular, will be the exact eigenstates of the Hubbard model. Anti-ferromagnetic phase at zero dopping and chiral superconductivity with doping are confirmed in the Hubbard model.