Topology and Interactions in a Frustrated Slab: Tuning from Weyl Semimetals to $\mathcal{C}>1$ Fractional Chern Insulators

We show that, quite generically, a [111] slab of spin-orbit coupled pyrochlore lattice exhibits surface states whose constant energy curves take the shape of Fermi arcs, localized to different surfaces depending on their quasimomentum. Remarkably, these persist independently of the existence of Weyl points in the bulk. Considering interacting electrons in slabs of finite thickness, we find a plethora of known fractional Chern insulating phases, to which we add the discovery of a new higher Chern number state which is likely a generalization of the Moore-Read fermionic fractional quantum Hall state. By contrast, in the three-dimensional limit, we argue for the absence of gapped states of the flat surface band due to a topologically protected coupling of the surface to gapless states in the bulk. We comment on generalizations as well as experimental perspectives in thin slabs of pyrochlore iridates.

Figure 1

The pyrochlore slab. The left panel shows the [111] pyrochlore slab with $N=4$ kagome layers (dark blue) separated by (yellow) sites of $N-1=3$ triangular layers. A practical labeling of the $4N-1=15$ sites in the unit cell and the basis vectors, ${\mathbf{a}}_1,{\mathbf{a}}_2$, of the Bravais lattice are also indicated. The top right panel indicates the considered nearest-neighbor processes. The lower right panel shows the local environment of a triangular (yellow) site for which the local constraint of destructive interference directly leads to the surface states (1).

Figure 2

Weakly coupled Chern insulators versus Weyl semimetals. As $t_{\perp }$ is increased there is a transition from a weakly coupled regime to a distinct phase where Weyl nodes occur on the line connecting $\mathrm{\Gamma }$ and $M$. In the top left panel we show the phase diagram in the case of nearest-neighbor hopping only (we set $t_1=-1$ throughout) [39]. The other panels show example band structures with fixed ${\lambda }_1=0.5$ and varying $t_{\perp }=1,2,3$ for a slab with $N=300$ kagome layers along the crucial $\mathrm{\Gamma }-K-M-\mathrm{\Gamma }$ path through the projected 2D Brillouin zone (BZ) (cf. top right inset). Note that, remarkably, the band highlighted in orange corresponding to the surface states (2), is independent of $t_{\perp }$.

Figure 3

Surface state structure and Fermi arcs. The color scale indicates the inverse penetration depth, ${\xi }^{-1}(\mathbf{k})=\log (|r(\mathbf{k})|)$ of the surface states throughout the 2D BZ for the same parameters, $t_1=-1,{\lambda }_1=0.5$, used in Fig. 2. The black lines illustrate Fermi arcs for a chemical potential set at the Weyl node for a few $t_{\perp }$ values. When ${\xi }^{-1}(\mathbf{k})$ changes sign, the localization changes between top (red) to bottom (blue) surfaces, hence splitting the Fermi circle into six spatially disjoint arcs.

Figure 4

From Weyl nodes to flat surface bands. For $t_1=-1,{\lambda }_1=0.3,{\lambda }_2=0.2,t_{\perp }=2.0$, we plot the energy dispersion for various $N$ and $t_2$ on the path $\mathrm{\Gamma }-K-M-\mathrm{\Gamma }$ through the BZ. In the left panel we set the number of kagome layers to $N=100$ while varying $t_2$. For $t_2=-0.3$ (top) there is a clearly visible Weyl node on the line connecting $M$ and $\mathrm{\Gamma }$. For $t_2=0$ the node is skewed and at $t_2=0.3$ it is essentially flattened while the surface band (bold orange) remains at almost fixed energy throughout the entire BZ. In the right panel we fix $t_2=0.3$ and reduce the number of layers, $N=30,10,5$, from top to bottom, and a sizable finite size gap quickly opens throughout the entire BZ. The band highlighted in bold orange is that of (2), carrying Chern number $\mathcal{C}=N$.

Figure 5

Topological degeneracy in the $\mathcal{C}=2$ bilayer system. (a) The energy spectra of the interaction $H=\sum _{⟨i,j,k⟩}{n}_i{n}_j{n}_k$ interaction projected to the flatband for $N_e=8$ and $N_e=10$ electrons in lattices with $N_1\times N_2=(N_e/2)\times 6$ unit cells yielding a filling fraction of $\nu =1/3$. Each energy level is labeled by the conserved many-body momentum ($K_1,K_2$). The shaded area indicates the nine quasidegenerate states. (b) The $y$-direction spectral flow for eight electrons under twisted boundary conditions $\mathrm{\Psi }({\mathbf{r}}_j+N_2{\mathbf{a}}_2)=\exp (i{\mathrm{\Phi }}_2)\mathrm{\Psi }({\mathbf{r}}_j)$ of the ground state $\mathrm{\Psi }({\mathbf{r}}_j)$. The red, green, and blue dots represent the nine quasidegenerate states in different momentum sectors, and the gray dots represent the excited states. The parameters are $t_1=-1,{\lambda }_1=0.9,t_2={\lambda }_2=0$.

Figure 6

Ground state occupation numbers. $n(\mathbf{k})$ plotted against $|\xi [r(\mathbf{k})]{|}^{-1}=|\log (|r(\mathbf{k})|)|$ (left panel) and $-E_h(\mathbf{k})$ (right panel, cf. Ref. [41]) for Chern numbers $\mathcal{C}=2$ (blue diamond) and $\mathcal{C}=100$ (red square) at $\nu =1/3$. This illustrates a general trend: $n(\mathbf{k})$ is inhomogenous at large $\mathcal{C}$, while it can remain comparably constant for small $\mathcal{C}$ (and $\nu$) [39].