Understanding physics of high-$T_c$ cuprate superconductors remains one of the important problems in materials science. Though a number of diverse theories argue about the superconductivity and competing orders, ab initio and quantitative understanding is lacking. Here, we reproduce the experimental phase diagram of ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+y}$ by solving its ab initio low-energy effective Hamiltonian without adjustable parameters. It shows a superconducting phase in a wide range of hole density $\delta$, and its competition with charge period-4 plus spin period-8 stripe order near $\delta \sim 0.1$, in agreement with experimental results including recent x-ray scattering. Then a crucial role of off-site interactions in stabilizing the superconductivity is elucidated with emphasis on charge fluctuations. It also clarifies the condensation energy mainly contributed from the on-site Coulomb interaction. The present achievement will enable deeper, predictable understanding on open issues of the high-$T_c$ superconducting mechanism and promote ab initio studies on strongly correlated electrons beyond parametrized model studies.

• Received 31 January 2019
• Revised 28 April 2019

DOI:https://doi.org/10.1103/PhysRevB.101.045124