Exact Out-of-Time-Ordered Correlators in Free Luttinger Liquids: Theory and Perturbative Corrections

In a free Luttinger liquid, we calculate the exact out-of-time-ordered correlator (OTOC) for vertex operators, which is valid for arbitrary operator charges without small-charge approximations. The OTOC grows purely as a power law ~ t γ inside the light cone without exponential spreading, characteristic of integrable systems. We develop a systematic perturbative expansion in the interaction strength λ and construct the full structure of first- and second-order corrections via Wick’s theorem. At finite temperature T = 1/β, thermal effects modify the late-time scaling from power-law to exponential growth ~ exp(γπt/β); however, this reflects thermal statistics rather than quantum chaos. These results lay a solid foundation for studies of information scrambling and quantum chaos in one-dimensional integrable systems.