  Although flat-band structures have attracted intensive studies in condensed
matter and optical physics due to their eigenstates exhibiting huge degeneracy
and allowing for the localization of wave packet, it is not clear how the flat
band of Liouvillian influences the relaxation dynamics of open quantum systems.
To this end, we study the dynamical signatures of Liouvillian flat band in the
scheme of Lindblad master equation. Considering a chain model with gain and
loss, we demonstrate three kinds of band dispersion of Liouvillian: flat bland,
dispersionless only in the real part and imaginary part, and capture their
dynamical signatures: when the rapidity spectrum of Liouvillian is flat, the
particle numbers in different sites relax to its steady state value with the
same decay rate; when the real or imaginary part of rapidity spectrum is
dispersionless, the relaxation behaviors have oscillating or forked
characteristics. We also unveil that the Liouvillian flat band can lead to
dynamical localization, which is characterized by the halt of propagation of a
local perturbation on the steady state.
