Spontaneous fractional Chern insulators in transition metal dichalcogenide moiré superlattices

The Moiré superlattice realized in two-dimensional heterostructures offers an exciting platform to access strongly correlated electronic states. In this work, we study transition metal dichalcogenides (TMD) Moiré superlattices with time-reversal symmetry and nontrivial spin/valley-Chern numbers. Utilizing realistic material parameters and the method of exact diagonalization, we find that at a certain twisting angle and fractional filling, gapped fractional topological states, i.e., fractional Chern insulators, are naturally stabilized by simply introducing the Coulomb repulsion. In contrast to fractional quantum Hall systems, where the time-reversal symmetry has to be broken explicitly, these fractional states break the time-reversal symmetry spontaneously. We show that the Chern number contrasting in the opposite valleys imposes a strong constraint on the nature of fractional Chern insulator and the associated low-energy excitations.

Figure 1

(a) Schematic view of the Moiré superlattice. (b) We choose the Moiré Brillouin zone (MBZ) to be the rhombus and the origin in momentum space is chosen at $M$. (c) Moiré band structure at $\theta =1.{38}^{\circ }$. The top Moiré band is nearly flat with Chern number $\pm 1$. (d) The gap ratio $\frac{{\mathrm{\Delta }}_{12}}{W}=\frac{min\left[E_1\left(\mathit{k}\right)\right]-max\left[E_2\left(\mathit{k}\right)\right]}{max\left[E_1\left(\mathit{k}\right)\right]-min\left[E_1\left(\mathit{k}\right)\right]}$ as a function of twisted angle $\theta$, where $E_1\left(\mathit{k}\right)$ $\left[E_2\left(\mathit{k}\right)\right]$ is the energy of the first (second) topmost Moiré band.

Figure 2

Numerical diagonalization results for eight particles in $4\times 6$ Moiré lattice. We choose $\theta =1.{38}^{\circ }$ and $U=1.38\mathrm{meV}$. The bandwidth at this twist angle is $W=0.083\mathrm{meV}$. Here $N=N_1\times N_2/3$ is the number of particles. (a) Energy spectrum with three nearly degenerate ground states in each valley. (b) The occupation number of single particle states $n(k_1,k_2)$ for each of the three many-body ground states. The nearly uniform distribution of $n(k_1,k_2)$ suggests the ground state is an incompressible liquid. (c) Under flux insertion along $k_2$ direction, the ground states evolve into each other. (d) Particle entanglement spectrum (PES) for the separation of $N_A=4$ particles.

Figure 3

(a) The many-body gap $\mathrm{\Delta }$ for various system sizes at $v=1/3$ filling. The interaction strength is fixed to be $U=1.38\mathrm{meV}$. The increase of $\mathrm{\Delta }$ in $4\times 6$ system suggests the gap persists in the two-dimensional thermodynamic limit. (b) The phase diagram for Fermi liquid (FL), FL with valley polarization (VP) and fractional Chern insulator (FCI) at different interaction strength $U\left(\epsilon \right)=\frac{e^2}{4\pi \epsilon {\epsilon }_0{a}_M}$ and twisted angle $\theta$. The dashed line corresponds to $U\left(\epsilon \right)={\mathrm{\Delta }}_{12}$, above which the interaction starts to mix different bands and the single-band approximation breaks down.

Figure 4

Dispersion of valley wave excitation $E_w\left(\mathit{q}\right)$ for eight particles in $4\times 6$ lattice. Excitation above the ground state with total momentum $k_1=0,k_2=0$ ($k_1=0,k_2=2$) are labeled by the squares and circles, respectively. The slight energy difference for these two ground states is caused by finite-size effect. The inset compares the energy of the lowest valley wave excitation $E_w$, the lowest intravalley many-body excitation $E_{mb}$, and the ground-state energy $E_g$.

Figure 5

FCI at $v=2/5$. (a) The energy spectrum of eight particles in $4\times 5$ system, and (b) six particles in $3\times 5$ system. (c) The flux insertion for the system in (a), where the five ground states are marked in red (some of them are on top of each other). A finite gap remains during flux insertion. (d) The particle entanglement spectrum for (a) with $N_A=3$. There are $51\times 20=1020$ states below the dashed line, consistent with quasihole counting.