Quantum Monte Carlo study of the interplay between spin stripe order and superconductivity in a two-orbital Hubbard model of cuprates

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 employ large-scale, unbiased constraine<i>d</i>-path quantum Monte Carlo simulations based on a two-orbital Hubbard model for copper oxides. We investigate the influence of the <i>Cu</i> <i>d</i>_{3<i>z</i>²-<i>r</i>²} orbital on the superconducting properties and spin stripe order in two prototypical cuprates, LSCO and HBCO.<br>First, within square lattice models of sizes 8 × 8 and 16 × 16, we examine the effect of the <i>Cu</i> <i>d</i>_{3<i>z</i>²-<i>r</i>²} orbital on superconductivity. By exploiting the differences in orbital energy-level separations among different cuprate materials, our numerical results demonstrate that, compared with LSCO, HBCO exhibits a significantly stronger enhancement in both the pairing correlation function and the effective pairing correlation function associated with <i>d</i>-wave superconducting symmetry. This result indicates that the higher superconducting transition temperature of HBCO relative to LSCO is closely related to the role of the <i>Cu</i> <i>d</i>_{3<i>z</i>²-<i>r</i>²} orbital.<br>Second, considering that 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, conventional periodic boundary conditions cannot accurately capture the intrinsic anisotropy of spin stripe order. To overcome this limitation, rectangular lattices are employed in our numerical simulations to describe the spin stripe configurations. This choice allows multiple stripe periods to be accommodated along the transverse direction, thereby faithfully capturing the spontaneously formed spin stripe structures in the electronic spin distribution and enabling a reliable analysis of their interplay with superconductivity. Based on this approach, we investigate the formation of spin stripe order in LSCO and HBCO using a 16 × 4 rectangular lattice. The numerical results show that LSCO develops relatively long single-domain spin stripes, whereas HBCO exhibits periodic spin stripe structures consisting of multiple domains. These findings indicate that LSCO hosts locally ordered spin stripes, while HBCO supports nonlocal, long-range ordered spin stripe order. More importantly, the pairing correlation and effective pairing correlation functions associated with <i>d</i>-wave superconductivity in HBCO retain a pronounced 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.<br>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 <i>Cu</i> <i>d</i>_{3<i>z</i>²-<i>r</i>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.