Vortexability: A unifying criterion for ideal fractional Chern insulators

Fractional Chern insulators realize the remarkable physics of the fractional quantum Hall effect (FQHE) in crystalline systems with Chern bands. The lowest Landau level (LLL) is known to host the FQHE, but not all Chern bands are suitable for realizing fractional Chern insulators (FCI). Previous approaches to stabilizing FCIs focused on mimicking the LLL through momentum-space criteria. Here, instead, we take a real-space perspective by introducing the notion of vortexability. Vortexable Chern bands admit a fixed operator that introduces vortices into any band wavefunction while keeping the state entirely within the same band. Vortexable bands admit trial wavefunctions for FCI states, akin to Laughlin states. In the absence of dispersion and for sufficiently short-ranged interactions, these FCI states are the ground state—independent of the distribution of Berry curvature. Vortexable bands are much more general than the LLL, and we showcase a recipe for constructing them. We exhibit diverse examples in graphene-based systems with or without magnetic field, and with any Chern number. A special class of vortexable bands is shown to be equivalent to the momentum-space “trace condition” or “ideal band condition”. In addition, we also identify a more general form of vortexability that goes beyond this criterion. We introduce a modified measure that quantifies deviations from general vortexability, which can be applied to generic Chern bands to identify promising FCI platforms.

Figure 1

(Left) A vortexable band (2) with associated projector $\mathcal{P}$ and complement $\mathcal{Q}=I-\mathcal{P}$. Acting with $\widehat{\mathsf{z}}$ keeps the wavefunction within the vortexable band: $\widehat{\mathsf{z}}\psi =(\mathcal{P}+\mathcal{Q})\widehat{\mathsf{z}}\psi =\mathcal{P}\widehat{\mathsf{z}}\psi$. (Right) the vortex metric $g\left({\mathit{r}}_0\right)$ from Eq. (6) describes the shape of the dip in probability density induced by the vortex attachment (3), (4).

Figure 2

The “magic line” of chiral twisted graphene. Bandwidth of chiral TBG, Eq. (17), as a (a) discrete or (b) continuous function of internal strain and angle. We take a realistic [84] internal strain vector potential and modulate its strength as ${\alpha }_s\mathcal{A}\left(\mathit{r}\right)$, where ${\alpha }_s$ interpolates from 0 to the realistic value of 1. Details in the text.

Figure 3

Flowchart describing examples, equivalent conditions, and consequences of vortexability. Several models, all describable using continuum chiral Hamiltonians (11), lead to vortexable bands. Such bands satisfy the trace condition with a suitable choice of periodic wavefunction (33) if the band is rotationally symmetric; if not a further linear transformation (37) may be required. With this choice, and after a suitable nonunitary gauge transformation, the Bloch wavefunctions $u^{\prime }$ can be chosen to be holomorphic in $k_x+i{k}_y$. Topologically nontrivial vortexable bands lead to SRI-GS: FQHE ground states in the limit of short-range interactions.

Figure 4

Internal strain of TBG. (a) Displacement field $\mathit{u}\left(\mathit{r}\right)$ in units of the graphene scale $a_{\mathrm{Gr}}$. The solid hexagon is a real-space unit cell. (b) Magnitude of the displacement field $\left|\mathit{u}\right(\mathit{r}\left)\right|$. (c) Psuedomagnetic field in real space. All data is for $\theta =1.{05}^{\circ }$ and uses ab initio strain parameters from [84].