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The out-of-time-ordered correlator (OTOC) is central to the understanding ofinformation scrambling in quantum many-body systems. In this work, we show thatthe OTOC in a quantum many-body system close to its critical point obeysdynamical scaling laws which are specified by a few universal criticalexponents of the quantum critical point. Such scaling laws of the OTOC imply auniversal form for the butterfly velocity of a chaotic system in the quantumcritical region and allow one to locate the quantum critical point and extractall universal critical exponents of the quantum phase transitions. Wenumerically confirm the universality of the butterfly velocity in a chaoticmodel, namely the transverse axial next-nearest-neighbor Ising model, and showthe feasibility of extracting the critical properties of quantum phasetransitions from OTOC using the Lipkin-Meshkov-Glick (LMG) model.

Dynamical Scaling Laws of Out-of-Time-Ordered Correlators