The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the
d-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration ($\ensuremath{\delta}$) dependence of the Hubbard model in the ground states on a square lattice at strong coupling
U/t=10, for the on-site interaction
U and the transfer
t, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of
0.01t. We have identified distinct phases including a uniform
d-wave superconducting phase for $0.17\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.22$ and a stripe charge/spin ordered phase for $\ensuremath{\delta}\ensuremath{\lesssim}0.17$ with the stripe period depending on $\ensuremath{\delta}$, together with presumable spatially coexisting antiferromagnetic and stripe order for $\ensuremath{\delta}\ensuremath{\lesssim}0.07$ and coexisting stripe and
d-wave superconductivity for $0.07\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.17$. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around $\ensuremath{\delta}=0.17$ in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.
The two-dimensional Hubbard model was originally proposed as the minimal model of uniform
d-wave superconductivity in the cuprate high-temperature superconductors, but recent calculations with relevant parameters in a realistic doping regime have often found other ground states. Further, uncertainty remains over such results due to the intrinsic bias of different computational methods. Here we address this uncertainty through state-of-the-art neural tensor network and neural backflow state simulations, enabled by large-scale GPU computation, that together reach an unprecedented variational accuracy on lattices with up to 1024 sites. At
1/8 doping in the pure Hubbard model, as well as with next-nearest-neighbor hopping relevant to the hole-doped cuprates, we reveal a near degeneracy among a low-energy uniform
d-wave state, superconducting stripes, and non-superconducting stripe states on the milli-
t energy scale. Our work rebalances the narrative that the Hubbard model supports a single dominant order in relevant parameter regimes towards a picture of extreme competition between stripes and superconductivity.
We study the competition between stripe states with different periods and a uniform
d-wave superconducting state in the extended two-dimensional Hubbard model at 1/8 hole doping using infinite projected entangled-pair states (iPEPS). With increasing strength of negative next-nearest neighbor hopping ${t}^{\ensuremath{'}}$, the preferred period of the stripe decreases. For the values of ${t}^{\ensuremath{'}}$ predicted for cuprate high-
Tc superconductors, we find stripes with a period 4 in the charge order, in agreement with experiments. Superconductivity in the period 4 stripe is suppressed at
1/8 doping. Only at larger doping, $0.18\ensuremath{\lesssim}\ensuremath{\delta}<0.25$, the period 4 stripe exhibits coexisting
d-wave superconducting order. The uniform
d-wave state is only favored for sufficiently large positive ${t}^{\ensuremath{'}}$.
The doped Hubbard model is a simple model for high-
Tc cuprate superconductors, while its ground state remains a challenge. Here, by performing state-of-the-art variational Monte Carlo calculations for the strong-coupling Hubbard model, we find evidence that the
d-wave superconducting phase emerges always near the phase separation region and the superconducting order has one-to-one correspondence with the enhancement of charge compressibility. The order as well as the phase separation are vulnerable to realistic intersite Coulomb interaction, while the superexchange interaction enhances both. An appropriate combination of these two widens the stable superconducting phase.
Inhomogeneities on various length scales suggesting phase separation seem to be a generic feature of hole-doped high-temperature superconducting cuprates. Also firmly established is the d-wave symmetry of the superconducting state. The interplay of superconductivity and phase separation has been noted in previous mean-field studies of correlated electron models. To further explore this issue in a more rigorous approach we have conducted quantum Monte Carlo calculations of a two-dimensional Hubbard model with nearest-neighbor attractive interaction. For a vanishing on-site repulsion, the result agrees with that of Micnas, Ranninger, and Robaszkiewicz [Phys. Rev. B 39, 11 653 (1989)], and indicates that for a given strength of the attractive interaction, homogeneous d-wave superconducting phase can exist only above a critical doping concentration, below which phase separation occurs. A finite on-site repulsion modifies the phase separation behavior. A stronger d-wave pairing correlation is seen in the inhomogeneous phase compared to the homogeneous phase.
Recent experimental results of angle-resolved photoemission spectroscopy suggested that an additional strong nearest neighbor attraction in the Hubbard model might be significant for the doped cuprates. The stripe-ordered patterns, which are formed by the inhomogeneous distribution of spin, charge, and pairing correlations in
CuO2 planes, are a well-known feature of the doped cuprates. By employing constrained path quantum Monte Carlo, we examine the effect of the nearest neighbor attraction on a two-dimensional repulsive Hubbard model. Within and across the stripe regions, the ground-state spin correlations and the
d-wave pairing correlation are calculated. We found that the spin-spin correlation is the largest when the interstripe region is nearly half filled, and we also found the
d-wave superconducting correlation of the neighboring sites is enhanced in the presence of the stripe pattern. This reveals the crucial effects of the strong nearest neighbor attraction on superconductivity in doped cuprates.