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We numerically analyze superconductivity (SC) in the cuprate superconductorsby using ab initio effective Hamiltonians consisting of the antibondingcombination of Cu $3d_{x^2-y^2}$ and O $2p_{\sigma}$ orbitals. We performvariational Monte Carlo calculations for the four carrier doped cuprates withdiverse experimental optimal SC critical temperature $T_{c}^{\rm opt}$:CaCuO$_2$ ($T_{c}^{\rm opt} \sim 110$ K), Bi$_2$Sr$_2$CuO$_6$ ($T_{c}^{\rm opt}\sim 10$-$40$ K), Bi$_2$Sr$_2$CaCu$_2$O$_8$ ($T_{c}^{\rm opt} \sim 85$-$100$K), and HgBa$_2$CuO$_4$ ($T_{c}^{\rm opt} \sim 90$ K). Materials and holedoping concentration ($\delta$) dependencies of the SC order parameter $F_{\rmSC}$ and the competition with spin/charge order show essential and quantitativeagreements with the available experiments in the following points: (1) Theground state is commonly the SC state, which is severely competing with thecharge/spin stripe and antiferromagnetic states. (2) $F_{\rm SC}$ showsamplitude consistent with the superfluid density measured in the muon spinresonance and its dome structure found in $\delta$ dependence shows consistencywith that of the SC gap in the tunneling and photoemission measurements. Wefurther find insights into the universal SC mechanism: (I) $F_{\rm SC}$increases with the ratio $U/|t_1|$, indicating that $U/|t_1|$ is the principalcomponent controlling the SC. Here, $U$ and $t_1$ are the onsite Coulombrepulsion and the nearest neighbor hopping, respectively, in the Hamiltonians.(II) A universal scaling $T_{c}^{\rm opt}\sim 0.16 \lvert t_1 \rvert F_{\rmSC}$ holds. (III) SC is enhanced and optimized if $U$ is increased beyond thereal available materials. It is further enhanced by decreasing the offsiteinteraction. The present findings provide useful clues for the design of new SCmaterials with even higher $T_{c}^{\rm opt}$.

physical review. x

Superconductivity studied by solving ab initio low-energy effective  Hamiltonians for carrier doped CaCuO$_2$, Bi$_2$Sr$_2$CuO$_6$,  Bi$_2$Sr$_2$CaCu$_2$O$_8$, and HgBa$_2$CuO$_4$