  Finding guiding principles to optimize properties of quantum anomalous Hall
(QAH) insulators is of pivotal importance to fundamental science and
applications. Here, we build a first-principles QAH material database of
chirality and band gap, explore microscopic mechanisms determining the QAH
material properties, and obtain a general physical picture that can
comprehensively understand the QAH data. Our results reveal that the usually
neglected Coulomb exchange is unexpectedly strong in a large class of QAH
materials, which is the key to resolve experimental puzzles. Moreover, we
identify simple indicators for property evaluation and suggest material design
strategies to control QAH chirality and gap by tuning cooperative or competing
contributions via magnetic co-doping, heterostructuring, spin-orbit proximity,
etc. The work is valuable to future research of magnetic topological physics
and materials.
