Effects of Doping on Stripe Order and Superconductivity in Cuprates: A Variational Monte Carlo Approach

High-temperature superconductors constitute a very special class of materials, whose puzzling properties and rich physics have caught the attention of the physicists soon after their discovery. The peculiar self-organization of the electrons into periodic structures of wavelength λ breaking the periodicities of the underlying lattice, called stripes, is found to coexist with superconducting correlations. By employing Variational Monte Carlo simulations on a single-band Hubbard model on the square lattice with both nearest-(t) and next-nearest-(t′) neighbour hopping, the present work investigates the consequences of increasing hole doping on the instauration of stripes and the behavior of the superconducting order parameter, with a particular focus on cuprate superconductors, along with a discussion on how the two phenomena affect each other. We consider two different values of the next-nearest neighbour hopping parameter, that are appropriate for describing cuprate superconductors. We observe that stripes are the optimal state in a wide doping range, while they melt into a uniform state for large values of the doping. A larger absolute value of the next-nearest neighbour hopping is found to penalize the formation of stripes. The existence of charge and spin order, associated to the static structure factor N(q) and the spin-spin correlations S(q) respectively, is discussed, along with the metallic or insulating character of the stripes. Superconducting pair-pair correlations D(x) indicating the presence of superconductivity are, on the other hand, suppressed with respect to the uniform state, suggesting a competing interplay between the two. Our findings shed light on the underlying physics of these complex materials and contribute to a deeper understanding of high-temperature superconductivity, with potential applications in the development of new and improved superconducting technologies.