  Quantum many-body systems are expected to relax to a thermal state over time,
with some exceptions such as systems with atypical eigenstates. In this study,
we investigate the effect of the existence of spatially localized eigenstates
on the relaxation dynamics of interacting bosons loaded into a one-dimensional
sawtooth lattice, which exhibits a flat band in the single-particle spectrum by
tuning the hopping rates. Using the time-evolving block decimation algorithm,
we simulate the time evolution of the local density profile based on the
Bose-Hubbard model with different initial conditions. Our results show the
presence of the flat band leads to a significant slowing down of relaxation for
weak interactions. Even for strong interaction, when the initial state includes
an isolated localized single-particle eigenstate in the superposition, remnants
of the initial bias in the density profile persist for a long time. This
particular relaxation dynamics can be tested using ultracold atoms in optical
lattices.
