Fractional Quantum Hall Effect of Hard-Core Bosons in Topological Flat Bands

Recent proposals of topological flat band models have provided a new route to realize the fractional quantum Hall effect without Landau levels. We study hard-core bosons with short-range interactions in two representative topological flat band models, one of which is the well-known Haldane model (but with different parameters). We demonstrate that fractional quantum Hall states emerge with signatures of an even number of quasidegenerate ground states on a torus and a robust spectrum gap separating these states from the higher energy spectrum. We also establish quantum phase diagrams for the filling factor $1/2$ and illustrate quantum phase transitions to other competing symmetry-breaking phases.

Figure 1

(a) The Haldane model on the honeycomb lattice and (b) the checkerboard model. The arrow directions present the signs of the phases $\pm \varphi$ in the NNN or NN hopping terms. For the checkerboard lattice, the NNN hopping amplitudes are $t^{\prime }$ ($-t^{\prime }$) along the solid (dotted) lines. The NNNN hoppings are represented by the dashed curves in both models. (c)–(d) Intensity plots of spectrum gaps in the $V_1\mathrm{\text{−}}{V}_2$ phase space at $\nu =1/2$ for (c) a 24-site honeycomb lattice and (d) a 24-site checkerboard lattice. FQHE, SF, SS1/SS2, and Solid label estimated phase regions inferred from the spectrum-gap plots and other information (see the text).

Figure 2

$1/2$ FQHE spectrum gaps versus $1/N_s$: (a)–(b) honeycomb lattice; (c)–(d) checkerboard lattice.

Figure 3

(a)–(b) Low energy spectra versus ${\theta }_1$ at a fixed ${\theta }_2=0$ for two honeycomb lattices at $\nu =1/2$. (c) $F({\theta }_1,{\theta }_2)\Delta {\theta }_1\Delta {\theta }_2/2\pi$ at $10\times 10$ mesh points for a GSM of the 32-site honeycomb lattice.

Figure 4

32-site honeycomb lattice at $\nu =1/2$. (a) $S^{AA}(\mathbf{q})$ of the FQHE phase; (b) $S^{AB}(\mathbf{q})$ of the SF phase; (c) $S^{AB}(\mathbf{q})$ of the solid phase. (d) Excited energy $E_n-E_1$ in various sectors (with $d$-fold degeneracy) versus $V_2$ at $V_1=4.0$. (e) ${\rho }_{\mathrm{SF}}$, $S^{AB}(\pi ,\pi )$, GS overlap $|⟨\Psi (V_2)|\Psi (10)⟩|$, and fidelity susceptibility ${\chi }_F$ versus $V_2$, at a fixed $V_1=4.0$. (f) Illustration of boson occupancy in the solid phase.

Figure 5

32-site checkerboard lattice at $\nu =1/2$. (a) $S^{AA}(\mathbf{q})$ of the FQHE phase. (b) $S^{AB}(\mathbf{q})$ of the SS1 phase; (c) $S^{AA}(\mathbf{q})$ of the SS2 phase. (d) ${\rho }_{\mathrm{SF}}$, $S^{AA}(\pi ,\pi )$, and ${\chi }_F$ versus $V_1$ along the $V_1=V_2$ line. (e)–(f) Illustration of boson occupancy in the SS1 and SS2 phases, respectively.

Figure 6

The $1/4$ FQHE of the 40-site checkerboard model: (a) spectrum gaps $E_n\mathrm{\text{−}}{E}_1$ versus $k_1{N}_2+k_2$; (b) low energy spectra versus ${\theta }_1$ at a fixed ${\theta }_2=0$.