A Variational Improvement of the Hartree-Fock Approach to the 2D Hubbard Model

We consider a refinement of the usual Hartree-Fock method applied to the 2D Hubbard model, in Nambu spinor formulation. The new element is the addition of a “condensate inducing” term proportional to a variational parameter h to the Hartree-Fock Hamiltonian, which generates an s- or d-wave condensate at zero temperature. This modified Hartree-Fock Hamiltonian is used only to generate variational trial states; energy expectation values are computed in the full two-dimensional Hubbard Hamiltonian with no modification. It is found that there exist trial states with non-vanishing condensates which are lower in energy than the standard Hartree-Fock ground states. However, these lower energy condensate states exist only in a spatially inhomogeneous (stripe) phase. No lowering of energy, relative to the Hartree-Fock ground state, is found in the spatially homogenous region of the U-density phase plane.