Superconducting and charge-ordered states in the anisotropic t–J–U model

Abstract Motivated by the effect of symmetry breaking in cuprates superconductors YBa $$_2$$ 2 Cu $$_3$$ 3 O $$_{7-\delta }$$ 7 - δ , we employ the renormalized mean-field theory to study the presence of uniform superconducting and charge-ordered states in two anisotropic t–J–U models, either with hopping strength anisotropy or antiferromagnetic interaction anisotropy. In the case of uniform superconducting state, compared with the isotropic t–J–U model with only $$d_{x^2-y^2}$$ d x 2 - y 2 -wave superconducting state, there is an additional s-wave superconducting state in the model with hopping strength anisotropy. Meanwhile, the hopping anisotropy may enhance the critical Coulomb interaction $$U_c$$ U c at the Mott insulator to the Gossamer superconductor transition point, and strong hopping anisotropy may weaken the superconducting state. In the case of a charge-ordered state, hopping anisotropy may suppress the amplitude of the charge density waves and pair density waves, which originate from local Coulomb interactions. These results indicate that the effects of hopping anisotropy and local Coulomb interactions are competitive. Moreover, the antiferromagnetic interaction anisotropy only weakly suppresses the superconducting gap and density wave amplitude. Our results show that the t–J–U model with hopping anisotropy is qualitatively consistent with experimental superconducting pair symmetry and charge density waves in the YBa $$_2$$ 2 Cu $$_3$$ 3 O $$_{7-\delta }$$ 7 - δ system.