Exotic Non-Abelian Topological Defects in Lattice Fractional Quantum Hall States

We investigate extrinsic wormholelike twist defects that effectively increase the genus of space in lattice versions of multicomponent fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a “proof of concept” demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.

Figure 1

Our model is equivalent to two square lattice layers (blue and red) where each plaquette is pierced by an effective flux $\varphi$ (upper left panel). We only plot nearest-neighbor hopping for simplicity. Defects are introduced through branch cuts (transparent gray) where the particles switch layer (green). We study systems with up to two such branch cuts, corresponding to topologies resembling wormholes, as displayed in the bottom panels.

Figure 2

Band structure for a $L_x\times L_y=12\times 12$ lattice with $\varphi =1/2$. (a) The spectrum $\{{\epsilon }_n\}$ of $H_0$. Without defects ($M=0$), ${\epsilon }_1,\dots ,{\epsilon }_{144}$ are exactly degenerate at zero energy. With one branch cut [$M=1$, white dashed line in (b)] at $(5.5,2.5\to 8.5)$, the original band structure is distorted, with two nearly degenerate clusters $({\epsilon }_1,{\epsilon }_2)$ and $({\epsilon }_{144},{\epsilon }_{145})$ having the largest deviation. (b) The lattice site weight of eigenvectors ${\psi }_1$, ${\psi }_2$, ${\psi }_{144}$, ${\psi }_{145}$ of $H_0$ for the same defects as in (a). All of them are strongly localized near the defects. The eigenstates with less energy deviation from the original band structure, for example, ${\psi }_3$, ${\psi }_4$, ${\psi }_{142}$, ${\psi }_{143}$, are less localized (not shown here). (c) The spectrum $\{{\epsilon }_n^R\}$ of $H_0+V$ with $R=0$, 1, and 2 and the same defects as in (a). ${\epsilon }_1^R,\dots ,{\epsilon }_{145}^R$ which we must flatten (shaded in gray) becomes more degenerate for larger $R$, with the flatness $({\epsilon }_{2\varphi L_x{L}_y+M+1}^R-{\epsilon }_{2\varphi L_x{L}_y+M}^R)/({\epsilon }_{2\varphi L_x{L}_y+M}^R-{\epsilon }_1^R)\approx 0.6,3.1,9.4$ for $R=0$, 1, 2.

Figure 3

Defect-enhanced topological degeneracy for Abelian systems at $\nu =1/2$ with two branch cuts [35]. (a) The energy spectra of various system sizes. The eight quasidegenerate ground states are highlighted by the cyan shade. (b) The $y$-direction spectral flow for $N_b=6$, $L_x\times L_y=4\times 3$, $\varphi =1/2$. The eight ground states (blue $+$) never mix with excited states (gray $▵$). (c) The PES (blue) for $N_b=8$, $L_x\times L_y=6\times 4$, $\varphi =1/3$ in the $N_b^A=4$ sector and the corresponding quasihole excitations (red) for $N_b=4$, $L_x\times L_y=6\times 4$, $\varphi =1/3$. The number of states below the gaps (indicated by the gray $\}$) are identical in both spectra.

Figure 4

Defect-enhanced topological degeneracy for non-Abelian systems. The approximately degenerate ground states, together with the degeneracy $\mathcal{D}$, are highlighted by the cyan shade. (a) The energy spectra at $\nu =1$ with one pair of defects [35]. (b) The energy spectra at $\nu =1$ with two pairs of defects [35]. (c) The $y$-direction spectral flow for $N_b=10$, $L_x\times L_y=3\times 5$, $\varphi =1/3$ with the same branch cut as in (a). (d) The $y$-direction spectral flow for $N_b=10$, $L_x\times L_y=3\times 5$, $\varphi =1/3$ with the same branch cuts as in (b). (e) The energy spectrum at $\nu =3/2$ with one pair of defects [35].