Floquet-Bloch theory: a periodically driven band E₀(k) = −2t·cos(k) spawns photon replicas at E₀(k) ± nΩ.
Coupling A opens avoided crossings where replicas cross the bare band. The effective hopping is renormalized:
t_eff = t · J₀(A/Ω) — Bessel function renormalization (Haldane-Floquet topological transitions at zeros of J₀).
Colors indicate Floquet band index n; gap size ∝ A. The Floquet quasi-energy lies in [−Ω/2, Ω/2].