Does the Hubbard Model Show $d_{x^2-y^2}$ Superconductivity?

TL;DR

This paper rigorously proves that the two-dimensional Hubbard model does not exhibit $d_{x^2-y^2}$-wave superconducting long-range order at any temperature, challenging assumptions about its role in high-temperature superconductivity.

Contribution

It provides a rigorous proof that the 2D Hubbard model cannot have $d_{x^2-y^2}$-wave long-range order at any nonzero temperature or zero temperature with an energy gap.

Findings

No $d_{x^2-y^2}$-wave order at nonzero temperature

Order excluded at zero temperature if charge gap opens

Results align with quantum Monte Carlo simulations

Abstract

It is rigorously shown that the two-dimensional Hubbard model with narrow bands (including next nearest-neighbor hopping, etc.) does not exhibit $d_{x^2 -y^2}$-wave pairing long-range order at any nonzero temperature. This kind of pairing long-range order will also be excluded at zero temperature if an excited energy gap opens in the charge excitation spectrum of the system. These results hold true for both repulsive and attractive Coulomb interactions and for any electron fillings, and are consistent with quantum Monto Carlo calculations.