Band geometry of fractional topological insulators

Recent numerical simulations of flat-band models with interactions which show clear evidence of fractionalized topological phases in the absence of a net magnetic field have generated a great deal of interest. We provide an explanation for these observations by showing that the physics of these systems is the same as that of conventional fractional quantum Hall phases in the lowest Landau level under certain ideal conditions which can be specified in terms of the Berry curvature and the Fubini-Study or quantum metric of the topological band. In particular, we show that when these ideal conditions hold, the density operators projected to the topological band obey the $W_{\infty }$ algebra. Our approach leads us to propose a way of testing the suitability of topological bands for hosting fractionalized phases.

Figure 1

The hexagonal unit cell for a tight-binding model on the honeycomb lattice with nontrivial Chern insulator phases. $t_1$ and $t_2{e}^{i\pm \varphi }$ are the nearest- and next-nearest-neighbor hopping amplitudes. The model also includes a term that gives on-site energies of $M$ and $-M$ to the two triangular sublattices.

Figure 2

Contour plots of the distribution of Berry curvature over a unit cell of the reciprocal lattice; this unit cell was chosen so as to not split up the Brillouin-zone corners, which appear to be points of high Berry curvature. The same color scale is used in each panel.

Figure 3

Characteristic ellipsoids corresponding to the quantum metric, i.e., the region satisfied by $\mathbf{x}·g·\mathbf{x}<1$. Colors are taken from the Berry curvature.

Figure 4

Contour plots of $\text{tr}\left[g\right]-\left|B\right|$, i.e., the difference between the trace of the quantum metric and the absolute value of the Berry curvature, for the same parameter values.