Superconducting instabilities of the non-half-filled Hubbard model in two dimensions

The problem of weakly correlated electrons on a square lattice is formulated in terms of a one-loop renormalization group. Starting from the action for the entire Brillouin zone we reduce successively the cutoff Λ about the Fermi surface and follow the renormalization of the coupling U as a function of three energy-momenta. We calculate the intrinsic temperature scale ${\mathit{T}}_{\mathit{co}}$ where the renormalization-group flow crosses over from the regime (Λ>${\mathit{T}}_{\mathit{co}}$) where the electron-electron (e-e) and electron-hole (e-h) terms are equally important to the regime (Λ${\mathit{T}}_{\mathit{co}}$) where only the e-e term plays a role. In the low-energy regime only the pairing interaction V is marginally relevant, containing contributions from all renormalization-group steps of the regime Λ>${\mathit{T}}_{\mathit{co}}$. We identify the most attractive eigenvalue ${\lambda }_{\min }$ of ${\mathit{V}}_{\mathrm{\Lambda }={\mathit{T}}_{\mathit{co}}}$. At low filling, ${\lambda }_{\min }$ corresponds to the ${\mathit{B}}_2$ representation (${\mathit{d}}_{\mathit{xy}}$ symmetry), while near half filling the strongest attraction occurs in the ${\mathit{B}}_1$ representation (${\mathit{d}}_{{\mathit{x}}^2\mathrm{-}{\mathit{y}}^2}$ symmetry). In the direction of the van Hove singularities, the order parameter shows peaks with increasing strength as one approaches half filling. We also give a possible interpretation of angle-resolved photoemission spectroscopy experiments trying to determine the symmetry of the order parameter in the high-${\mathit{T}}_{\mathit{c}}$ compound ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_8$. © 1996 The American Physical Society.