  Quench dynamics of topological phases have been studied in the past few years
and dynamical topological invariants are formulated in different ways. Yet most
of these invariants are limited to minimal systems in which Hamiltonians are
expanded by Gamma matrices. Here we generalize the dynamical 3-winding-number
in two-band systems into the one in generic multi-band Chern insulators and
prove that its value is equal to the difference of Chern numbers between
post-quench and pre-quench Hamiltonians. Moreover we obtain an expression of
this dynamical 3-winding-number represented by gapless fermions in phase bands
depending only on the phase and its projectors, so it is generic for the quench
of all multi-band Chern insulators. Besides, we obtain a multifold fermion in
the phase band in (k, t) space by quenching a three-band model, which cannot
happen for two band models.
