Quantum anomalous hall effect based on MnBi$_2$Te$_4$ -family systems

The quantum anomalous Hall effect (QAHE) enables dissipationless chiral edge transport in the absence of external magnetic fields, driven by the interplay of non-trivial band topology and spontaneous ferromagnetism. Initially discovered in magnetically doped topological insulators, the intrinsic disorder in these systems restricts the observable temperature to the millikelvin regime. The identification of the intrinsic magnetic topological insulator MnBi2Te4 (MBT) has marked a significant advancement in the field, providing a stoichiometric platform characterized by ordered magnetic sublattices. In this review, we present a comprehensive overview of the progress within the MBTfamily, connecting notable experimental breakthroughs with emerging theoretical predictions. We trace the evolution from doped systems to intrinsic MBT, critically analyzing layer-dependent magnetic properties, the realization of high-Chern-number states, and recent advancements in molecular beam epitaxy growth and surface passivation engineering that have facilitated precise quantization. Simultaneously, we systematically summarize the diverse spectrum of theoretical and computational studies that extend beyond conventional topology. We emphasize recent predictions related to intrinsic ferroelectricity, odd-parity magnetism, and light-induced quantum optical phenomena, such as Floquet engineering, which provide innovative pathways to manipulate topological order. Finally, we offer an outlook on future directions, including the exploration of homologous series heterostructures and the interplay between MBT-based physics and fractionalized topological states.