  The adiabatic edge state pumping (ESP) in one dimensional model, which has
important applications in topological phase transition and the associated
implementation of edge states in quantum simulation, has been widely performed
in both theories and experiments. This phenomenon has been verified in some
small physical models, yet some fundamental issues about this process have not
been clarified. In this paper, we revisit this problem of ESP and pinpoint a
pair of non-adiabatic points in the band levels, at which the adiabatic
condition breaks down. We determine the two points using the criteria of
non-adiabaticity. As a result, the oscillation of ESP as evolution time varies
can be resolved in terms of Landau-Zener-St\"{u}ckelberg (LZS) interference.
Furthermore, in the presence of disorder, we show that the ESP may break down
for the anticrossing between the edge and the bulk levels, where the
non-adiabaticity diverges. Thus in a relatively long chain with weak disorder,
we demonstrate the failure of the ESP. This new type of ESP unveiled in this
work is readily accessible in experiment, and shall therefore lead to a
down-to-earth platform for the intriguing LZS dynamics.
