Dynamically screened Coulomb interaction in the parent compounds of hole-doped cuprates, trends and exceptions

Although the cuprate high-temperature superconductors were discovered already 1986 the origin of the pairing mechanism remains elusive. While the doped compounds are superconducting with high transition temperatures $T_{c}$ the undoped compounds are insulating due to the strong effective Coulomb interaction between the Cu $3d$ holes. We investigate the dependence of the maximum superconducting transition temperature, $T_{c\text{ max}}$, on the onsite effective Coulomb interaction $U$ using the constrained random-phase approximation. We focus on the commonly used one-band model of the cuprates, including only the antibonding combination of the Cu $d_{x^2-y^2}$ and O $p_x$ and $p_y$ orbitals, and find a clear screening dependent trend between the static value of $U$ and $T_{c\text{ max}}$ for the parent compounds of a large number of hole-doped cuprates. Our results suggest that superconductivity is favored by a large onsite Coulomb repulsion. We analyze both the trend in the static value of $U$ and its frequency dependence in detail and, by comparing to other works, speculate on the mechanisms behind the trend.