Pairing in the Two-Dimensional Hubbard Model from Weak to Strong Coupling

TL;DR

This study investigates the evolution of superconducting instabilities in the 2D Hubbard model across different interaction strengths, demonstrating a smooth crossover from weak to strong coupling with consistent pairing hierarchies.

Contribution

It provides a comprehensive comparison of DCA, perturbative, and RPA methods, revealing adiabatic continuity and clarifying the limitations of RPA in dilute regimes.

Findings

Agreement of pairing hierarchies between methods near half-filling

Robust d_{x^2-y^2}-wave pairing at strong coupling

Discrepancy of RPA in dilute limit with p-wave pairing

Abstract

The Hubbard model is the simplest model that is believed to exhibit superconductivity arising from purely repulsive interactions, and has been extensively applied to explore a variety of unconventional superconducting systems. Here we study the evolution of the leading superconducting instabilities of the single-orbital Hubbard model on a two-dimensional square lattice as a function of onsite Coulomb repulsion $U$ and band filling by calculating the irreducible particle-particle scattering vertex obtained from dynamical cluster approximation (DCA) calculations, and compare the results to both perturbative Kohn-Luttinger (KL) theory as well as the widely used random phase approximation (RPA) spin-fluctuation pairing scheme. Near half-filling we find remarkable agreement of the hierarchy of the leading pairing states between these three methods, implying adiabatic continuity between weak- and strong-coupling pairing solutions of the Hubbard model. The $d_{x^2-y^2}$-wave instability is robust to increasing $U$ near half-filling as expected. Away from half filling, the predictions of KL and RPA at small $U$ for transitions to other pair states agree with DCA at intermediate $U$ as well as recent diagrammatic Monte Carlo calculations. RPA results fail only in the very dilute limit, where it yields a $d_{xy}$ ground state instead of a $p$-wave state established by diagrammatic Monte Carlo and low-order perturbative methods, as well as our DCA calculations. We discuss the origins of this discrepancy, highlighting the crucial role of the vertex corrections neglected in the RPA approach. Overall, comparison of the various methods over the entire phase diagram strongly suggests a smooth crossover of the superconducting interaction generated by local Hubbard interactions between weak and strong coupling.