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

Abstract

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 μ. 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.