Universally optimizable strategy for magnetic gaps towards high-temperature quantum anomalous Hall states via magnetic-insulator/topological-insulator building blocks

Optimizing the magnetic Zeeman-splitting term, specifically the magnetic gap of the topological surface states (TSSs), is a crucial issue and central challenge in advancing higher-temperature quantum anomalous Hall (QAH) states. In this work, we demonstrate a counterintuitive, nonmonotonic relationship between the magnetic gap and the hybridization strength in ferromagnetic-insulator (FMI)/topological-insulator (TI) sandwich structures. Concretely, insufficient hybridization strength fails to induce a substantial magnetic gap; while excessive hybridization incandesces the competition between kinetic and Coulomb exchange interactions, thereby reducing the gap. Strong hybridization strength also spatially delocalizes the TSSs, diminishing the effective orbital coupling between TSS-based $p$ and magnetic $d$ orbitals, which further weakens kinetic and Coulomb exchange interaction strength. Moreover, modifying the stacking order offers an experimentally viable approach to optimizing magnetic gaps, enabling the tunability of Chern numbers, chirality, and maximizing global gaps. These findings unveil a universal guiding principle for optimizing magnetic gaps in FMI-TI proximity-based QAH systems, offering valuable insights for advancing experimental implementations in this field.