We propose a method for controlling the exchange interactions of Mott insulators with strong spin-orbit coupling. We consider a multiorbital system with strong spin-orbit coupling and a circularly polarized light field and derive its effective Hamiltonian in the strong-interaction limit. Applying this theory to a minimal model of $\alpha \text{−}{\mathrm{RuCl}}_3$, we show that the magnitudes and signs of three exchange interactions, $J$, $K$, and $\mathrm{\Gamma }$, can be changed simultaneously. Then, considering another case in which one of the hopping integrals has a different value and the other parameters are the same as those for $\alpha \text{−}{\mathrm{RuCl}}_3$, we show that the Heisenberg interaction $J$ can be made much smaller than the anisotropic exchange interactions $K$ and $\mathrm{\Gamma }$.

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• Exchange interaction
• Light-induced magnetic effects
• Spin-orbit coupling
• Floquet systems
• Honeycomb lattice
• Mott insulators
• Bloch-Floquet theorem
• Spin lattice models
• Tight-binding model