  We find that a flat-band fermion system with interactions and without
disorders exhibits non-thermalized ergodicity-breaking dynamics, an analog of
many-body localization (MBL). In the previous works, we observed flat-band
many-body localization (FMBL) in the Creutz ladder model. The origin of FMBL is
a compact localized state governed by local integrals of motion (LIOMs), which
are to be obtained explicitly. In this work, we clarify the dynamical aspects
of FMBL. We first study dynamics of two-particles, and find that the states are
not substantially modified by weak interactions, but the periodic time
evolution of entanglement entropy emerges as a result of a specific mechanism
inherent in the system. On the other hand, as the strength of the interactions
is increased, the modification of the states takes place with inducing
instability of the LIOMs. Furthermore, many-body dynamics of the system at
finite fillings is numerically investigated by time-evolving block decimation
(TEBD) method. For a suitable choice of the filling, non-thermal and low
entangled dynamics appears. This behavior is a typical example of the
disorder-free FMBL.
