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.
Highlights
Since the theoretical proposal by Kohn and Luttinger (KL) [1,2] that superconductivity can arise from purely repulsive electron interactions and the subsequent discovery of superconductivity in materials like heavy fermions, cuprates, organic Bechgaard salts, and iron-based superconductors, superconducting instabilities in models of interacting fermions have been extensively studied
In contrast to dynamical cluster approximation (DCA), both the KL and random phase approximation (RPA) schemes are limited to small values of the Coulomb interaction of U = O(t ), and RPA is sensitive to the inherent magnetic instability that occurs upon increasing U
The number of gap nodes resolved at the Fermi level is sensitive to the cluster size; at U = 2 which is calculated for a cluster size of N = 64, twenty nodes are resolved at the Fermi surface for the third subleading dx2−y2 DCA solution
Summary
Since the theoretical proposal by Kohn and Luttinger (KL) [1,2] that superconductivity can arise from purely repulsive electron interactions and the subsequent discovery of superconductivity in materials like heavy fermions, cuprates, organic Bechgaard salts, and iron-based superconductors, superconducting instabilities in models of interacting fermions have been extensively studied. We propose to understand the evolution of the pairing interaction in the Hubbard model as a function of doping and correlation strength With this question in mind, we calculate the superconducting pairing vertex of the Hubbard model via numerical solutions of the Bethe-Salpeter equation obtained in the dynamical cluster approximation (DCA) with quantum Monte Carlo (QMC) impurity solver. This approximation [51,52] is known to provide an accurate estimate of the pairing vertex over a range of intermediate strength U values appropriate for the cuprates [53].
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