Adiabatic Continuation of Fractional Chern Insulators to Fractional Quantum Hall States

We show how the phases of interacting particles in topological flat bands, known as fractional Chern insulators, can be adiabatically connected to incompressible fractional quantum Hall liquids in the lowest Landau level of an externally applied magnetic field. Unlike previous evidence suggesting the similarity of these systems, our approach enables a formal proof of the equality of their topological orders, and furthermore this proof robustly extends to the thermodynamic limit. We achieve this result using the hybrid Wannier orbital basis proposed by Qi [Phys. Rev. Lett. 107, 126803 (2011)] in order to construct interpolation Hamiltonians that provide continuous deformations between the two models. We illustrate the validity of our approach for the ground state of bosons in the half filled Chern band of the Haldane model, showing that it is adiabatically connected to the $\nu =1/2$ Laughlin state of bosons in the continuum fractional quantum Hall problem.

Figure 1

(a) Geometry and hopping terms in the Haldane model: The fundamental unit cell has two inequivalent sites $A$ and $B$. Second nearest neighbor interactions are complex, with arrows indicating the direction of a positive hopping phase ${\varphi }_{\mathbf{r}{\mathbf{r}}^{\prime }}$. (b) Berry curvature for the Haldane model. (c) Corresponding expectation value of the position operator $⟨{\widehat{X}}^{cg}⟩$ for Wannier states according to Eq. (1).

Figure 2

(a) Magnitudes of matrix elements for the two-body delta interaction between particle pairs of incoming (outgoing) momenta $J_{12}^{i(f)}=[(J_1+J_2)\mathrm{mod}{N}_{\mathrm{cell}}]$ in a finite size geometry for the FQHE on the torus with $N_{\varphi }=12$ (left) and for the FCI in the lowest band of the Haldane model for a $3\times 4$ lattice (right). (b) Schematic showing some short range interactions, including a “squeezing” process $V_{11}$ as well as two diagonal interaction terms $V_{20}$, $V_{00}$. Panel (c) shows how the magnitude of these processes depends on the center of mass position for the FCI (solid) as compared to FQH (dashed).

Figure 3

(a) Overlap of the ground state manifolds of $\mathcal{H}(\kappa )$ and ${\mathcal{H}}^{\mathrm{FQH}}$ (see text) for bosons at $\nu =1/2$. (b) Total weight in torus ground state subspace. (c) Spectrum for a system with $N=10$ particles, along a path adiabatically connecting a continuum problem on the torus to the FCI Haldane model on a lattice of $4\times 5$ unit cells. The insets show the momentum-resolved spectrum for the torus (left) and the pure FCI system (right). (d) Gap for several systems of different sizes and aspect ratios. Inset: Finite size scaling of the gap.

Figure 4

Entanglement spectrum (a) and entanglement gap (b) for the reduced density matrix of a block with $N_A=N/2$ particles. System sizes in analogy with the energy spectrum shown in Figs. 3c, 3d.