Phase diagram of the t-U^2 Hamiltonian of the weak coupling Hubbard model

TL;DR

This paper investigates the phase diagram and symmetry of Cooper pairs in the weak coupling Hubbard model on a square lattice, revealing the dominance of d-wave pairing across various parameters and its dependence on the van Hove singularity.

Contribution

The study provides a detailed phase diagram and demonstrates the stability of d-wave pairing in the weak coupling regime for different lattice configurations and parameters.

Findings

d-wave pairing is dominant over a wide range of parameters

superconducting gap depends on the van Hove singularity position

d-wave pairing remains stable with anisotropic transfer t'

Abstract

We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction $U$, for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small $U$ is mapped onto an effective Hamiltonian with the attractive interaction using the canonical transformation: $H_{eff}=e^S He^{-S}$. The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling $n_e$ and the next nearest-neighbor transfer $t'$. The d-wave pairing is dominant for the square lattice in a wide range of $n_e$ and $t'$. The d-wave pairing is also stable for the square lattice with anisotropic $t'$. The three-band $d$-$p$ model is also investigated, for which the d-wave pairing is stable in a wide range of $n_e$ and $t_{pp}$ (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter $-t'$ could not be so large so that the optimum doping rate is in the range of $0.8 <n_e <0.9$.