Loop unitary and phase band topological invariant in generic multi-band Chern insulators

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.