Operator Scrambling and Fermi's Golden Rule*

The out-of-time-order correlator (OTOC) qualifies the scrambling of local operators over the entire system. It has been argued that at early times the OTOC exhibits an exponential growth with a rate bounded above by 2π/β, where β is the inverse temperature.

In this work, we show that for generic (0+1)D systems the OTOC is ultimately related to the thermal average of the Loschmidt echo, with the perturbation given by the coupling between subsystems. It is further argued that the exponential growth and the temperature dependence of the OTOC can be determined by the Fermi’s golden rule.