  We study quench dynamics in the many-body Hilbert space using two isolated
systems with a finite number of interacting particles: a paradigmatic model of
randomly interacting bosons and a dynamical (clean) model of interacting
spins-$1/2$. For both systems in the region of strong quantum chaos, the number
of components of the evolving wave function, defined through the number of
principal components $N_{pc}$ (or participation ratio), was recently found to
increase exponentially fast in time [Phys. Rev. E 99, 010101R (2019)]. Here, we
ask whether the out-of-time ordered correlator (OTOC), which is nowadays widely
used to quantify instability in quantum systems, can manifest analogous
time-dependence. We show that $N_{pc}$ can be formally expressed as the inverse
of the sum of all OTOC's for projection operators. While none of the individual
projection-OTOC's shows an exponential behavior, their sum decreases
exponentially fast in time. The comparison between the behavior of the OTOC
with that of the $N_{pc}$ helps us better understand wave packet dynamics in
the many-body Hilbert space, in close connection with the problems of
thermalization and information scrambling.

Statistical Mechanics