Flat bands with higher Chern number in pyrochlore slabs

A large number of recent works point to the emergence of intriguing analogs of fractional quantum Hall states in lattice models due to effective interactions in nearly flat bands with Chern number $C=1$. Here, we provide an intuitive and efficient construction of almost dispersionless bands with higher Chern numbers. Inspired by the physics of quantum Hall multilayers and pyrochlore-based transition-metal oxides, we study a tight-binding model describing spin-orbit coupled electrons in $N$ parallel kagome layers connected by apical sites forming $N-1$ intermediate triangular layers (as in the pyrochlore lattice). For each $N$, we find finite regions in parameter space giving a virtually flat band with $C=N$. We analytically express the states within these topological bands in terms of single-layer states and thereby explicitly demonstrate that the $C=N$ wave functions have an appealing structure in which layer index and translations in reciprocal space are intricately coupled. This provides a promising arena for new collective states of matter.

Figure 1

The kagome bilayer (a) and trilayer (b). The layers are colored differently for clarity and the considered hopping processes are indicated by arrows in (a). A 2D projection of the trilayer is shown in (c) and illustrates the general structure of the pyrochlore lattice: The projection has a threefold periodicity (kagome layer $m+3n$ has the same projection as layer $m$).

Figure 2

Bulk dispersion, $E_s\left(\mathbf{k}\right)$, $s=1,2,...,4N-1$, including topological flat bands in the presence of spin-orbit coupling. (a) shows a high-symmetry path through the Brillouin zone along which (b)–(i) show the dispersion of (1) for a system with $N=1,2,3,4,8,12,50,100$ stacked kagome layers, respectively. In each case there is a very flat band (bold orange line) with Chern number $C=N$. The parameters are chosen as $t_1=-1,t_2={\lambda }_1=0.3,{\lambda }_2=0.2,t_{\perp }=1.3$. As described in the text, the value of $t_{\perp }$ is not affecting the flat band but can be tuned to move the other bands (thin blue).

Figure 3

The flatness ratio $\Delta /W$ (defined in the text) as a function of $t_{\perp }$ for various $N$ (a). Notably, the flatness reaches its maximum for relatively large $t_{\perp }$. Here we use $t_1=-1,t_2=0.3,{\lambda }_1=0.348,{\lambda }_2=0.172$. The kink reflects that the upper bands become closer to the topological band than the lower bands. In (b) we plot $\Delta /W$ for $N=2$ kagome layers as a function of ${\lambda }_1$ and ${\lambda }_2$ for $t_{\perp }=1.3,t_1=-1,t_2=0.3$. The flatness ratio is large for a sizable region and, in fact, positive throughout the plotted range.

Figure 4

The probability distribution, $P_m\left(\mathbf{k}\right)$, of being in different kagome layers, ${\mathcal{K}}_m$, in the $C=N$ topological bands ($t_{\perp }=1.3,t_1=-1,t_2=0.3,{\lambda }_1=0.3,{\lambda }_2=0.2$). In (a)–(c) we display the probability distributions for each layer within the $C=2,3,4$ bands respectively through a high-symmetry path through the Brillouin zone [cf. Fig. 2a]. That the wave functions are completely localized to the kagome layers is manifested by ${\sum }_m{P}_m\left(\mathbf{k}\right)=1,\forall \mathbf{k}$. The wave function is entirely localized to the top or bottom kagome layer at the $K$ points. In (d) we display the total weight, $\langle P_m\rangle$, in each kagome layer for various $N$. Notably, most weight is carried by the outermost layers.