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The long-studied Hubbard model is one of the simplest models of copper-oxidesuperconductors. However, the connection between the model and the experimentalphase diagram is still under debate, in particular regarding the existence andextent of the $d$-wave superconducting phase. Recent rapid progress inimproving the accuracy of numerical solvers has opened a way to answer thisquestion reliably. Here, we study the hole-doping concentration ($\delta$)dependence of the Hubbard model in the ground states on a square lattice atstrong coupling $U/t=10$, for the on-site interaction $U$ and the transfer $t$,using a variational Monte Carlo method. The method, which combines tensornetwork and Lanczos methods on top of Pfaffian wave functions, reveals a richphase diagram, in which many orders compete severely and degenerate within theenergy range of 0.01$t$. We have identified distinct phases including a uniform$d$-wave superconducting phase for $0.17\lesssim \delta \lesssim0.22$ and astripe charge/spin ordered phase for $\delta\lesssim0.17$ with the stripeperiod depending on $\delta$, together with presumable spatially coexistingantiferromagnetic and stripe order for $\delta\lesssim0.07$ and coexistingstripe and $d$-wave superconductivity for $0.07\lesssim\delta\lesssim0.17$. Thepresent, improved method revealed a wider region of a charge uniformsuperconducting phase than the previous studies and shows a qualitativesimilarity to the phase diagram of the cuprate superconductors. Thesuperconducting order parameter is largest at doping of around $\delta=0.17$ inthe ground state, which undergoes phase transitions from an inhomogeneous to auniform state.

physical review. b./physical review. b

Stripe and superconducting order competing in the Hubbard model on a  square lattice studied by a combined variational Monte Carlo and tensor  network method