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

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