Arbitrarily high Chern numbers in spin-orbit coupled systems

Topological phases with high Chern numbers (HCNs) have garnered significant interest in research on the quantum anomalous Hall effect. However, topological states with Chern numbers $C>2$ remain scarce in both theoretical and experimental studies. In this paper, we introduce a mechanism for achieving arbitrary Chern numbers in spin-orbit coupled systems. This mechanism leverages phase-controlled Floquet engineering to induce high-order spin flips. Notably, the realization of unbounded HCNs is contingent upon the phase relationship in the temporally modulated spin-flip coefficient. Furthermore, to implement this mechanism, we propose a self-designed dual-set optical Raman lattice that drives atomic transitions, thereby demonstrating a quantum simulator capable of accessing the $C\ensuremath{\ge}5$ regime. Our findings not only overcome the current limits on achievable Chern numbers, but also open alternative avenues for exploring different topological phases, with potential applications in quantum technologies such as topological quantum computation and quantum simulation.