Floquet Topological Phases

Floquet theorem: under periodic driving H(t+T)=H(t), the time evolution U(T)=T·exp(-i∫₀ᵀH dt) defines quasi-energies ε ∈ (-π/T, π/T] (the Floquet Brillouin zone). The Floquet SSH alternates hoppings: H₁=J₁Σσˣ for t∈[0,T/2], H₂=J₂Σσˣ for t∈[T/2,T]. New topological invariants arise: besides ν=1 modes at ε=0 (like static SSH), there can be π-modes at ε=π/T — edge states oscillating at the drive frequency, with no equilibrium analog. The full classification: (ν₀, νπ) = (winding at 0, winding at π). Anomalous phase: ν₀=0, νπ=1 (only possible with drive).