  We study operator dynamics in many-body quantum systems, focusing on generic
features of systems which are ergodic, spatially extended, and lack conserved
densities, as exemplified by spin chains with Floquet time evolution. To
characterise dynamics we examine, in solvable models and numerically, the
behaviour of operator autocorrelation functions, as a function of time and the
size of the operator support. The standard expectation is that operator
autocorrelation functions in such systems are maximum at time zero and decay,
over a few Floquet periods, to a fluctuating value that reduces to zero under
an average over an ensemble of statistically similar systems. Our central
result is that ensemble-averaged correlation functions also display a second
generic feature, which consists of a peak at a later time. In individual
many-body systems, this peak can also be revealed by averaging autocorrelation
functions over complete sets of operators supported within a finite spatial
region, thereby generating a partial spectral form factor. The duration of the
peak grows indefinitely with the size of the operator support, and its
amplitude shrinks, but both are essentially independent of system size provided
this is sufficiently large to contain the operator. In finite systems, the
averaged correlation functions also show a further feature at still later
times, which is a counterpart to the so-called ramp and plateau of the spectral
form factor; its amplitude in the autocorrelation function decreases to zero
with increasing system size. Both the later-time peak and the ramp-and-plateau
feature are specific to models with time-translation symmetry, such as Floquet
systems or models with a time-independent Hamiltonian, and are absent in models
with an evolution operator that is a random function of time, such as the
extensively-studied random unitary circuits.
