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The usual Hartree-Fock approximation to the Hubbard model is based oneigenstates of the electron number operator. But this formulation is notunique. A different (and inequivalent) version, formulated in terms of Nambutwo-component spinors, is based on eigenstates of the difference between thenumbers of spin up and spin down electrons, with electron density dependent onparameters $U,t,t'$ and chemical potential $\mu$. The advantage of thisformulation is that electron pairing condensates can be directly computed. Weshow that in the ground states away from half-filling, obtained in this "Nambu"Hartree-Fock approximation, a discrete rotation symmetry is spontaneouslybroken, and the condensates exhibit the expected d-wave form in momentum space.We also show that the Mott insulator electron configuration is obtained in thisformulation at large $U/t$ and half-filling, and roughly locate the boundary,in the hole doping$-$$U/t$ plane, between a region of local antiferromagnetismand stripe/domain formation, and the region of d-wave condensation.

Hartree-Fock with Nambu spinors, and d-wave condensation in the 2D Hubbard model