  We investigate the orthogonality catastrophe and quantum speed limit in the
Creutz model for dynamical quantum phase transitions. We demonstrate that exact
zeros of the Loschmidt echo can exist in finite-size systems for specific
discrete values. We highlight the role of the zero-energy mode when analyzing
quench dynamics near the critical point. We also examine the behavior of the
time for the first exact zeros of the Loschmidt echo and the corresponding
quantum speed limit time as the system size increases. While the bound is not
tight, it can be attributed to the scaling properties of the band gap and
energy variance with respect to system size. As such, we establish a relation
between the orthogonality catastrophe and quantum speed limit by referencing
the full form of the Loschmidt echo. Significantly, we find the possibility of
using the quantum speed limit to detect the critical point of a static quantum
phase transition, along with a decrease in the amplitude of noise induced
quantum speed limit.
