Microscopic origin of the highest-T_c among cuprate superconductors recorded in a tri-layer HgBa₂Ca₂Cu₃O₈

We investigate the superconducting properties of a tri-layer cuprate superconductor HgBa₂Ca₂Cu₃O₈ (Hg1223) at ambient pressure and under pressure, motivated by its highest superconducting critical temperature among the cuprates. Using variational Monte Carlo simulations combined with machine learning techniques, we analyze the ab initio low-energy effective Hamiltonian of Hg1223. Our results demonstrate self-doping resulting in a distinct behavior of carrier density between the inner and outer CuO₂ planes, leading to differing superconducting order parameters across the layers. The dome structure of T_c observed in pressure dependence is quantitatively reproduced by applying the relation between T_c at the optimum doping and the calculated superconducting order parameter F_SC established in single- and double-layer cuprates. These findings enable to extract the microscopic mechanisms that drive high-T_c in multi-layer cuprates and provide valuable insights into the design and optimization of future superconducting materials.