  Exotic topological states of matter such as Floquet topological insulator or
Floquet Weyl semimetal can be induced by periodic driving. This work proposes a
Floquet semimetal with Floquet-band holonomy. That is, the system is gapless,
but as a periodic parameter viewed as quasimomentum of a synthetic dimension
completes an adiabatic cycle, each Floquet band as a whole exchanges with other
Floquet bands. The dynamical manifestations of such Floquet-band holonomy are
studied. Under open boundary conditions, we discover anomalous chiral edge
modes localized only at one edge, winding around the entire quasienergy
Brillouin zone, well separated from bulk states, and possessing holonomy
different from bulk states. These remarkable properties are further exploited
to realize quantized or half-quantized edge state pumping.
