Superconductivity in the two-dimensional Hubbard model based on the exact pair potential

TL;DR

This paper provides an exact analytical study of superconductivity in the 2D Hubbard model, revealing anisotropic gap solutions and estimating critical temperatures that align with experimental data for cuprates.

Contribution

It introduces an exact calculation of the pair potential and solves the gap equation analytically, offering new insights into the anisotropic superconducting gap in the Hubbard model.

Findings

Gap is largest in the anti-nodal direction

Critical temperature Tc aligns with experimental values for cuprates

Superconducting gap persists up to high doping levels

Abstract

We analyze solutions to a superconducting gap equation based on the two-dimensional Hubbard model with nearest and next-to-nearest neighbor hopping. The Cooper pair potential can be calculated exactly and expressed in terms of elliptic functions. The Fermi surfaces at finite temperature and chemical potential are calculated based on the exact two-body S-matrix of the Hubbard model using the formalism we recently developed, which allows variation of hole doping. The resulting solutions to the gap equation are strongly anisotropic, namely largest in the anti-nodal direction, and zero in the nodal directions of the Brillouin zone. For U/t = 13 and t' /t =-0.3, appropriate to BSCO, and a physically natural choice for the cut-off, our self-contained analytic calculations yield the gap in the anti-nodal direction Delta/t = 0.06 and a maximum Tc/t = 0.04 at hole doping h=0.15. For phenomenological fits to the Fermi surfaces for cuprates, we obtain the comparable value Tc/t = 0.03 at optimal doping, both in good agreement with experiments. The superconducting gap is non-zero for all hole-doping h < 0.35, and increases all the way down to zero doping, suggesting that it evolves into the pseudogap.