Out-of-Time-Order Correlators

Non-time-ordered observables diagnosing commutator growth under quantum dynamics. The main concepts are OTOCs, operator spreading, Lieb–Robinson bounds. Each must be defined in the context of a concrete physical model, mathematical representation, or experimental task. Terminology alone does not establish the behavior of the system.

Out-of-time-order correlators probe how initially localized operators spread under Heisenberg time evolution. For unitary bounded operators, a squared commutator between W(t) and V can grow as their supports begin to overlap, reflecting information propagation. Exponential-like early growth in suitable large systems can be associated with scrambling, but finite small chains can display similar curves without a well-defined classical Lyapunov exponent.

Compute an exact finite-spin-chain OTOC with fixed W,V, ensemble, Hamiltonian and boundaries. Check the short-time expansion, Lieb–Robinson light-cone constraints and long-time finite-size saturation. Compare integrable and chaotic parameter choices while avoiding claims of a universal black-hole exponent from a narrow fitting interval.