We consider states of BCS form in the two-dimensional (2D) Hubbard model which, starting from some arbitrary point in state space in the neighborhood of a Hartree-Fock ground state, are relaxed within that BCS ansatz to local minima of the energy. As in the Hartree-Fock approximation, there are a vast number of local minima, nearly degenerate in energy. What is new and unlike the conventional Hartree-Fock states is that there is a region in parameter space where these local minima are clearly associated with $d$-wave condensates of the form $d_{x^2-y^2}$ in the underdoped region. There are also indications of $d_{xy}$ condensation in the overdoped region, at least in this approximation to the 2D Hubbard model, as well as condensates over a range of parameters on triangular lattices.

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• Superconducting order parameter
• d-wave
• Superconductors
• Unconventional superconductors
• BCS theory
• Variational approach