Fractional quantum Hall effect in topological flat bands with Chern number two

Recent theoretical works have demonstrated various robust Abelian and non-Abelian fractional topological phases in lattice models with topological flat bands carrying Chern number $C=1$. Here we study hard-core bosons and interacting fermions in a three-band triangular-lattice model with the lowest topological flat band of Chern number $C=2$. We find convincing numerical evidence of bosonic fractional quantum Hall effect at the $\nu =1/3$ filling characterized by threefold quasidegeneracy of ground states on a torus, a fractional Chern number for each ground state, a robust spectrum gap, and a gap in quasihole excitation spectrum. We also observe numerical evidence of a robust fermionic fractional quantum Hall effect for spinless fermions at the $\nu =1/5$ filling with short-range interactions.

Figure 1

(a) The three-band triangular-lattice model. The NN and NNN hopping amplitudes are positive (negative) along the solid (dashed) lines; the arrows represent the phases $\pm 2\phi$ (signs are represented by arrow directions) in the NN hoppings and $\pm \phi$ in the NNN hoppings. (b) Edge states of the triangular-lattice model in (a), and the lower TFB owns the Chern number $C=+2$.

Figure 2

The $1/3$ bosonic FQHE. (a) Low-energy spectrum $E_n-E_1$ versus the momentum $k_1{N}_2+k_2$ of the $1/3$ bosonic FQHE phase for three lattice sizes $N_s=45$, 54, and 63 at $\nu =1/3$ filling with $V_1=V_2=0.0$. (b) Spectrum gaps versus $1/N_s$ for four lattice sizes.

Figure 3

The $1/3$ bosonic FQHE. Low-energy spectra versus ${\theta }_2$ at a fixed ${\theta }_1=0$ for three lattice sizes at $\nu =1/3$ filling with $V_1=V_2=0.0$: (a) $N_s=36$; (b) $N_s=45$; (c) $N_s=54$.

Figure 4

The $1/3$ bosonic FQHE. Berry curvatures $F({\theta }_1,{\theta }_2)\Delta {\theta }_1\Delta {\theta }_2/2\pi$ at $10\times 10$ mesh points for the GSM of the $N_s=45$ lattice at $\nu =1/3$ filling with $V_1=V_2=0.0$: (a) the first GS in (0,0) sector; (b) the second GS in (1,0) sector; (c) the third GS in (2,0) sector.

Figure 5

The $1/3$ bosonic FQHE. Quasihole excitations for three lattice sizes with $V_1=V_2=0.0$: (a) $N_s=45$ and $N_b=4$; (b) $N_s=54$ and $N_b=5$; (c) $N_s=63$ and $N_b=6$.

Figure 6

The $1/5$ fermionic FQHE. Low-energy spectrum $E_n-E_1$ versus the momentum $k_1{N}_2+k_2$ of the $1/5$ fermionic FQHE for four lattice sizes with $V_1=8.0$ and $V_2=V_3=1.0$: (a) $N_s=45$; (b) $N_s=60$; (c) $N_s=75$; (d) $N_s=90$. The quasidegeneracy has been labeled for the (0,0) sector of the $N_s=75$ case in (c).

Figure 7

The $1/5$ fermionic FQHE. Low-energy spectra versus ${\theta }_1$ at a fixed ${\theta }_2=0$ for three lattice sizes at $\nu =1/5$ filling with $V_1=8.0$ and $V_2=V_3=1.0$: (a) $N_s=45$; (b) $N_s=60$; (c) $N_s=90$. The quasidegeneracy of the GSM has been labeled.