To clarify the origin of the distinct superconducting transition temperatures in cuprate high-temperature superconductors and to elucidate the relationship between spin stripe order and superconductivity, we conduct large-scale, unbiased constrained-path quantum Monte Carlo simulations based on a two-orbital Hubbard model for copper oxides. We investigate the influences of the Cu \mathrmd_3z^2-r^2 orbital on the superconducting properties and spin stripe order in two prototypical cuprates, LSCO and HBCO.

First, within the frame of square lattice models of sizes 8\times8 and 16\times16, we examine the effect of the Cu \mathrmd_3z^2-r^2 orbital on superconductivity, and simulate them by using the differences in orbital energy-level separations among different cuprate materials. The numerical results demonstrate that compared with LSCO, the HBCO exhibits that both the pairing correlation function and the effective pairing correlation function associated with d-wave superconducting symmetry are greatly enhanced. This result indicates that the higher superconducting transition temperature of HBCO than LSCO is closely related to the role of the Cu \mathrmd_3z^2-r^2 orbital.

Second, as spin stripe order spontaneously breaks the rotational symmetry of the lattice and forms unidirectional, periodically modulated spin-density structures whose periodicity is generally incompatible with that of a square lattice, traditional periodic boundary conditions cannot accurately capture the intrinsic anisotropy of spin stripe order. To overcome this limitation, we use rectangular lattices in our numerical simulations to describe the spin stripe configurations. This approach allows for accommodating multiple stripe periods along the transverse direction, thereby faithfully capturing the spontaneously formed spin stripe structures in the electronic spin distribution and enabling the reliable analysis of their interplay with superconductivity. According to this approach, we investigate the formation of spin stripe order in LSCO and HBCO by using a 16\times4 rectangular lattice. The numerical results show that the LSCO develops relatively long single-domain spin stripes, whereas the HBCO exhibits periodic spin stripe structures consisting of multiple domains. These findings indicate that the LSCO possesses locally ordered spin stripes, while the HBCO supports a nonlocal, long-range ordered spin stripe order. More importantly, the pairing correlation function and effective pairing correlation function related to d-wave superconductivity in HBCO maintain an obvious long-range enhancement, demonstrating that long-range ordered spin stripes are beneficial for enhancing superconductivity. This result reveals a cooperative interplay between spin stripe order and superconductivity.

Taken together, these results not only provide insight into the origin of the distinct superconducting transition temperatures in cuprate high-temperature superconductors and the correlations between different ordered phases, but also demonstrate that the Cu \mathrmd_3z^2-r^2 orbital plays a crucial role in tuning superconductivity and spin fluctuations in cuprate materials. Our study thus offers a new theoretical perspective for exploring strongly correlated cuprate systems.