Topological flat band models with arbitrary Chern numbers

We report the theoretical discovery of a systematic scheme to produce topological flat bands (TFBs) with arbitrary Chern numbers. We find that generically a multiorbital high Chern number TFB model can be constructed by considering multilayer Chern number $C=1$ TFB models with enhanced translational symmetry. A series of models are presented as examples, including a two-band model on a triangular lattice with a Chern number $C=3$ and an $N$-band square lattice model with $C=N$ for an arbitrary integer $N$. In all these models, the flatness ratio for the TFBs is larger than 30 and increases with increasing Chern number. In the presence of appropriate interparticle interactions, these models are likely to lead to the formation of Abelian and non-Abelian fractional Chern insulators. As a simple example, we test the $C=2$ model with hardcore bosons at $1/3$ filling, and an intriguing fractional quantum Hall state is observed.

Figure 1

The two-band model with Chern number $C=3$ on a triangular lattice. (a) Lattice structure and NN (solid bond) and NNNN (dashed curve) hoppings. The end point of a link is red (blue) if it is connected to an $A$ ($B$) orbital. (b) Lattice structure and NNN hoppings. Arrows show the sign of the phases. The green disk indicates that each unit cell contains one lattice site and the green rectangle represents the enlarged unit cell. (c) Single-particle energy spectrum as a function of $k_x$ and $k_y$. (d) Chiral edge states of the two-orbital triangular lattice model, where $k_1=k_x/2+\sqrt{3}{k}_y/2$. Each band has Chern number $C=\pm 3$.

Figure 2

(a)–(c) Trilayer Haldane model in an ABC-type trilayer structure. The green rectangle in each layer indicates the unit cell of the Haldane model. The dashed black circles mark the positions of some sites in the other two layers. (d) Chiral edge state of the Haldane model, where $k_1=k_x/2+\sqrt{3}{k}_y/2$.

Figure 3

(a), (b) The two-orbital Chern number two model on a square lattice. (a) NN and NNNN hoppings. (b) NNN hoppings. (c)–(e) The $1/3$ bosonic FQHE. (c) Low-energy spectrum $E_n-E_1$ versus the momentum $k_x{N}_y+k_y$ for three lattice sizes $N_s=18$, 24, and 48. (d) Low-energy spectrum as a function of boundary phase ${\theta }_x$ at a fixed ${\theta }_y=0$ at $1/3$ filling. (e) Spectrum gaps versus $1/N_s$ for three lattice sizes.

Figure 4

(a)–(d) Dispersions of the $N$-orbital square lattice model along high symmetry directions for $N=3$, 4, 8, and 20, where $t_1=1$, $t_2=-1/\sqrt{N}$, and $\varphi =\pi /N$. In each case the bottom band (red line) is very flat with Chern number $C=N$. (e) The flatness ratio of the bottom flat band increases exponentially with the number of orbitals $N$ (the Chern number $N$).