OTOCs & Quantum Chaos

OTOC C(t) = -⟨[W(t),V]²⟩ measures how a local perturbation V scrambles under time evolution. In chaotic systems, C(t) ~ e^{2λ_L t}/N (exponential scrambling) up to the scrambling time t* ~ (1/λ_L)log(N), followed by saturation. The MSS bound: λ_L ≤ 2π/β (saturated by black holes and SYK). For the kicked Ising model H = J·Σσᵢᶻσᵢ₊₁ᶻ + h·Σσᵢˣ: integrable (h=0) → Poisson level spacing; chaotic (h≠0, J≠0) → Wigner-Dyson (GUE/GOE). OTOC growth visualized as a "butterfly" spreading through the system.