Fractional Chern insulator with higher Chern number in optical lattice

Abstract

Fractional Chern insulators arise in topologically nontrivial flat bands, characterized by an integer Chern number C that corresponds to the number of dissipationless edge states in the non-interacting regime. Higher Chern numbers can replicate the physics of higher Landau levels and often confer enhanced topological robustness. However, realizing correlated fractional phases with higher Chern numbers in such flat band systems remains challenging. Here, we propose an interlayer coupling scheme to generate higher Chern numbers in a flat-band system, where the interlayer coupling transforms two C = 1 bands in a bilayer checkerboard lattice into a single flat band with C = 2 by lifting their degeneracy and merging their topological indices. Exact diagonalization calculation reveals that this engineered band hosts two fractional Chern insulator states with C = 2/3 and 2/5, respectively. An experimental setup is proposed to simulate these states using cold alkaline-earth-like atoms in an effective bilayer optical lattice. Our work provides a general and widely applicable strategy for constructing higher Chern number flat bands, opening a pathway to explore exotic fractional quantum phases

Figures (4)

• Figure 1: Experimental setup and energy bands.a Schematic diagram of the experimental setup for generating a bilayer optical lattice. The two central layers of the optical lattice are marked in red and blue, respectively. The large red and blue arrows surrounding the setup represent the corresponding laser beams from four distinct directions with different frequencies. The small arrows at the tails of the large arrows indicate the polarization of the lasers. b Illustration of the bilayer optical lattice, where $t_{\perp}$ denotes the interlayer hopping term. c Hopping terms within a single layer of the lattice. The NN hopping is represented by black solid lines with arrows, with strength $te^{\pm i\phi}$, where the direction of the arrows indicates the sign of the phase $\phi$. The NNN hopping between the A sublattice (red balls) and the B sublattice (blue balls) is denoted by dashed and dash-dotted lines, with strengths $t_1'$ and $t_2'$, respectively. The NNNN hopping terms are indicated by red and blue dashed lines, both with strength $t"$. d Energy level diagram of the alkaline-earth (-like) atoms used in the experimental system, illustrating the dependence of the atomic states on the relevant optical lattices. The wavelengths are $\lambda_1 = 790.02\text{ nm}$ and $\lambda_2 = 788.28 \text{ nm}$. e Band structure diagram of the system. Two flat bands are visible, and the Chern number of the second flat band is $2$. f Berry curvature plot under the same parameters as in d. g quantum metric plot under the same parameters as in d.
• Figure 2: The influence of interlayer coupling $t_{\perp}$, sizes and the phase diagram for $\nu_2 = 1/3$.a Low-energy spectrum at filling factor $\nu_2 = 1/3$ for electron numbers $N_e = 5, 8, 10$. b The ground state splitting energy and the energy gap for $N_x N_y=15,21,24,30$. c Depicts the low-energy spectra under identical parameters with varying interlayer hopping term $t_{\perp}$. When $t_{\perp}=1$, the band gap closes, and the system transitions from a FCI state to a trivial state. d Phase diagram as a function of the intralayer nearest-neighbor interaction $U$ and NNN interaction $U'$, with fixed interlayer hopping $t_{\perp}$ and filling factor $\nu_2=1/3$.
• Figure 3: Fractional Chern insulator state for $\nu_2=1/3$.a Spectral flow in the $y$-direction of the system at filling factor $\nu_2 = 1/3$ for $N_x = 4$, $N_y = 6$. b Excitation spectrum of the system for $N_x = 5$, $N_y = 5$, corresponding to the addition of a quasi-hole. There are 25 states below the energy gap (indicated by the black dashed line in the figure), uniformly distributed across each subspace. c Particle number distribution $n(\boldsymbol{k})$ in momentum space at the same parameters as in a. d The structure factor $S(\boldsymbol{q})$ at the same parameters as in a.
• Figure 4: Fractional Chern insulator state for $\nu_2 = 1/5$.a Low-energy spectrum of the system for $N_x = 5$, $N_y = 7$. b Spectral flow in the $x$-direction for $U = 2$, $U' = 4$. c Low-energy excitation spectrum of the system for $N_x = 6$, $N_y = 7$, corresponding to the addition of two quasi-holes. There are 189 states below the energy gap (indicated by the black dashed line in the figure). d Particle number distribution in momentum space of the system. e Corresponding structure factor $S(\boldsymbol{q})$.