  Scrambling of information in a quantum many-body system, quantified by the
out-of-time-ordered correlator (OTOC), is a key manifestation of quantum chaos.
A regime of exponential growth in the OTOC, characterized by a Lyapunov
exponent, has so far mostly been observed in systems with a high-dimensional
local Hilbert space and in weakly-coupled systems. Here, we propose a general
criterion for the existence of a well-defined regime of exponential growth of
the OTOC in spatially extended systems with local interactions. In such
systems, we show that a parametrically long period of exponential growth
requires the butterfly velocity to be much larger than the Lyapunov exponent
times a microscopic length scale, such as the lattice spacing. As an explicit
example, we study a random unitary circuit with tunable interactions. In this
model, we show that in the weakly interacting limit the above criterion is
satisfied, and there is a prolonged window of exponential growth. Our results
are based on numerical simulations of both Clifford and universal random
circuits supported by an analytical treatment.
