The Superconducting Instabilities of the non half-filled Hubbard Model in Two Dimensions

TL;DR

This paper uses a one-loop renormalization group approach to analyze superconducting instabilities in the non half-filled two-dimensional Hubbard model, identifying dominant pairing symmetries and their dependence on filling level.

Contribution

It introduces a comprehensive RG analysis starting from the full Brillouin zone to determine pairing symmetries and the intrinsic scale for superconductivity in the Hubbard model.

Findings

At low filling, the dominant pairing symmetry is d_{xy}

Near half filling, the dominant pairing symmetry is d_{x^2-y^2}

Order parameter peaks near van Hove singularities as half filling is approached

Abstract

The problem of weakly correlated electrons on a square lattice is formulated in terms of one-loop renormalization group. Starting from the action for the entire Brillouin zone (and not with a low-energy effective action) we reduce successively the cutoff $\Lambda$ about the Fermi surface and follow the renormalization of the coupling $U$ as a function of three energy-momenta. We calculate the intrinsic scale $T_{co}$ where the renormalization group flow crosses over from the regime ($\Lambda > T_{co}$) where the electron-electron (e-e) and electron-hole (e-h) terms are equally important to the regime ($\Lambda < T_{co}$) 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 $\Lambda > T_{co}$. After diagonalization of $V_{\Lambda =T_{co}}$, we identify its most attractive eigenvalue $\lambda _{\min}$. At low filling, $\lambda _{\min}$ corresponds to the $B_2$ representation ($d_{xy}$ symmetry), while near half filling the strongest attraction occurs in the $B_1$ representation ($d_{x^2-y^2}$ symmetry). In the direction of the van Hove singularities, the order parameter shows peaks with increasing strength as one approaches half filling. Using the form of pairing and the structure of the renormalization group equations in the low energy regime, we give our interpretation of ARPES experiments trying to determine the symmetry of the order parameter in the Bi2212 high-$T_{c}$ compound.