Loop unitary and phase band topological invariant in generic multiband Chern insulators

Quench dynamics of topological phases has 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 three-winding number in two-band systems into the one in generic multiband Chern insulators and prove that its value is equal to the difference of Chern numbers between postquench and prequench Hamiltonians. Moreover, we obtain an expression of this dynamical three-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 multiband Chern insulators. Besides, we obtain a multifold fermion in the phase band in $(\mathbf{k},t)$ space by quenching a three-band model, which cannot happen for two-band models.