  We explore quantum dynamics in Floquet many-body systems with local
conservation laws in one spatial dimension, focusing on sectors of the Hilbert
space which are highly polarized. We numerically compare the predicted charge
diffusion constants and quantum butterfly velocity of operator growth between
models of chaotic Floquet dynamics (with discrete spacetime translation
invariance) and random unitary circuits which vary both in space and time. We
find that for small but finite polarization per length (in the thermodynamic
limit), the random unitary circuit correctly predicts the scaling of the
butterfly velocity but incorrectly predicts the scaling of the diffusion
constant. We argue that this is a consequence of quantum coherence on short
time scales. Our work clarifies the settings in which random unitary circuits
provide correct physical predictions for non-random chaotic systems, and sheds
light into the origin of the slow down of the butterfly effect in highly
polarized systems or at low temperature.
