The honeycomb Fock-state lattice (FSL) constructed by the three-mode Jaynes-Cummings model has been proven powerful in simulating novel states of matter and engineering robust quantum devices. This setting is scalable in the number of coupled atoms with enriched physics. Here, we explore a dice FSL constructed by the Fock states in the three-mode Tavis-Cummings model where two atoms are symmetrically coupled to three cavity modes. By introducing Floquet-engineered phased hoppings between cavity modes, we construct the Haldane model on the dice FSL. This system exhibits quantum anomalous Hall bands with Chern numbers up to $\pm 2$ and hosts multiple analytically solvable chiral edge states. The dice-Haldane term can realize a spin-1 quantum circulator with photons routed by the collective atomic state, enabling the generation of multipartite entangled states. We also investigate the strain effects and the dynamics within the dice flat band that supports bulk circulating modes. These findings establish the dice FSL as a versatile and highly tunable platform for exploring topological phases of quantized light and engineering quantum devices with enhanced performance.

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• Cavity quantum electrodynamics
• Light-matter interaction
• Photonics