Floquet semimetal with Floquet-band holonomy

Exotic topological states of matter, such as a 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 the quasimomentum of a synthetic dimension completing 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.

Figure 1

Quasienergy spectrum of a Floquet semimetal, with $J=1$ and $V=-0.5$. Three Floquet bands can undergo interchange as $\lambda \to \lambda +2\pi$ adiabatically.

Figure 2

Pumping of a bulk eigenstate over three adiabatic cycles, with $J=1,V=-0.5$, and 150 lattice sites. (a) Quasienergy in ascending order for $\lambda =-\frac{\pi }{2}$ under periodic boundary condition. The $x$ axis represents the index of the states. The red star denotes the initial state, and the magenta cross or the green triangle denotes the state after one or two adiabatic cycles. Panels (b), (c), and (d) show sublattice populations for the initial state and the state after one and two adiabatic cycles, respectively.

Figure 3

Adiabatic pumping of Wannier states over three adiabatic cycles, with $J=1$ and $V=-0.5$. The wave-packet center prepared on the bottom, middle, and top Floquet bands in Fig. 1 at $\lambda =-\pi$ follows the red solid, green dashed, and blue dashed-dotted curves. Each adiabatic cycle lasts for 4000 driving periods.

Figure 4

Floquet spectrum under open-boundary conditions. The lattice contains 60 unit cells. The blue dots, red dashed line, and green dashed-dot line denote bulk states and left and right edge modes, respectively, with $J=1$. (a) $V=-0.9$, with two types of anomalous chiral edge modes marked by I and II; (b) $V=-0.5$; (c) $V=0$; and (d) $V=0.5$.

Figure 5

Edge-state pumping with edge states at different $\lambda$ [shown in Fig. 1] uniformly populated. In simulations, the open lattice has 150 lattice sites. Parts (a) and (b) show probability distribution $P$ of type-I edge modes and of the upper branch of type-II edge modes in the first 50 sites from the left. Parts (c) and (d) show the time dependence of pumped charge along the synthetic dimension for one driving period for type-I and type-II edge modes.