Topological flat Wannier-Stark bands

We analyze the spectrum and eigenstates of a quantum particle in a bipartite two-dimensional tight-binding dice network. In the absence of a dc bias, it hosts a chiral flatband with compact localized eigenstates. In the presence of a dc bias, the energy spectrum consists of a periodic repetition of one-dimensional energy band multiplets, with one member in the multiplet being strictly flat. The corresponding flatband eigenstates cease to be compact, and are localized exponentially perpendicular to the dc field direction, and superexponentially along the dc field direction. The band multiplets are characterized by a topological quantized winding number (Zak phase), which changes at specific values of the varied dc field strength. These changes are induced by gap closings between the flat and dispersive bands, and reflect the number of these closings.

Figure 1

(a) The dice lattice with the elementary cell consisting of three sites denoted by $A,B$, and $C$. The red and green circles represent the amplitudes $+1/\sqrt{6}$ and $-1/\sqrt{6}$, respectively, of a compact localized state. The black sites indicate an uncoupled trimer in the limit of infinitely strong dc bias. (b) Bloch bands of the dice lattice with a flatband at $E=0$ (for zero dc bias).

Figure 2

(a) WS band ladder of the biased dice lattice with $\mathbf{F}$ along the $y$ axis, $F=4$. Thick lines highlight the irreducible triplet of three WS bands. (b)–(d) Irreducible triplets for (b) $F=2$, (c) $F=1/0.62$, and (d) $F=1$.

Figure 3

The dual ladder system for the dice lattice: (a) tilted in the $y$ direction and (b) in the $x$ direction.

Figure 4

(a) Winding number of the phase $\chi$ as the function of the inverse field magnitude. (b) The gap between the dispersive bands of the irreducible triplet as a function of $1/F$ and $\vartheta$. In this representation, conical intersections and vanishing gaps of the dispersive bands with the parent flatband appear as black spots.

Figure 5

Integrated probabilities (a) $P\left(x\right)$ and (b) $P\left(y\right)$ for $F=4$. Inset: The localized state in the tilted dice lattice for $F=4$. Occupation amplitudes $\mathrm{\Psi }\left({\mathbf{R}}_j\right)$ of the lattice sites are shown as a color map with dark blue corresponding to $-1$ and bright yellow to $+1$.