Chern bands of twisted bilayer graphene: Fractional Chern insulators and spin phase transition

When one of the graphene layers of magic-angle twisted bilayer graphene is nearly aligned with its hexagonal boron nitride substrate (a configuration dubbed TBG/hBN), the active electronic bands are nearly flat and have a Chern number $C=\pm 1$. Recent experiments demonstrated a quantum anomalous Hall effect and spontaneous valley polarization at integer filling ${\nu }_T=3$ of the conduction band in this system. Motivated by this discovery, we ask whether fractional quantum anomalous Hall states (FQAHs) could also emerge in TBG/hBN. We focus on the range of filling fractions where valley ferromagnetism was observed experimentally. Using exact diagonalization, we find that the ground states at ${\nu }_T=\frac{10}{3}$ and ${\nu }_T=\frac{17}{5}$ are fractional Chern insulator states in the flatband limit (in the hole picture, these are the FQH fractions $\frac{2}{3}$ and $\frac{3}{5}$). The ground state is either spin polarized or a spin singlet, depending sensitively on band parameters. For nominally realistic band parameters, spin polarization is favored. Flattening the Berry curvature by changing a band parameter gives way to the spin singlet phase. Our estimation of the charge gap in the flatband limit shows that the FQAH state may be seen at accessible temperatures in experiments. We also study the effect of a nonzero bandwidth and show that there is a reasonable range of parameters in which the FQAH state is the ground state.

Figure 1

(a) Distribution of the Berry curvature density of the conduction band of TBG/hBN across the Brillouin zone for $w_0/w_1=0.0$ and $w_0/w_1=0.8$. (b) Evolution of the root mean square of the Berry curvature distribution with the relaxation parameter $w_0/w_1$, for two values of the twist angle $\theta =1.05$ and $\theta =1.15$. In the rest of the calculations, we fix $\theta =1.15$.

Figure 2

Numerical evidence for a FCI ferromagnet at ${\nu }_{h,T}=2/3$. (a) Low-energy spectrum of the system with $N=10$ fermions and $N_s=30$ moiré sites, for $w_0/w_1=0.8$, in the spin-polarized sector. (b) Finite-size extrapolation of the many-body gap $\mathrm{\Delta }E$ and ground state degeneracy splitting $\delta E$ for $w_0/w_1=0.8$.

Figure 3

(a) Low-energy spectrum of the system with $N=10$ spinful fermions and $N_s=15$ unit cells, for $w_0/w_1=0.0$, upon insertion of a magnetic flux $\phi$. The three topological sectors (red, green, and purple dots) have a spectral flow consistent with the topological order of the Halperin (112) state. Both the spin singlet states ($S=0$) and the other spin sectors are represented. (b), (c) Energy difference between the singlet ground state and the fully polarized ground state (b) as a function of the relaxation parameter $w_0/w_1$ (c) as a function of the root mean square $\tilde{F}$ of the Berry curvature distribution.

Figure 4

(a) Dispersion of the spin-wave excitation of the ferromagnetic ${\nu }_{h,T}=2/3$ FCI ground state at small $\mathbf{k}$. (b) Energy of the three types of particle-hole excitations. For $w_0/w_1<0.75$, the ground state is a spin singlet with a charge gap ${\mathrm{\Delta }}_E^{S=0}$. For $w_0/w_1\ge 0.75$, the ground state is ferromagnetic. $\mathrm{\Delta }E$ is the neutral gap of Fig. 2. The energy of a skyrmion particle-hole pair $E_{Sk}$ is extracted from a linear fit of ${\epsilon }_{\mathrm{SW}}\left(\mathbf{k}\right)$ at small $\mathbf{k}$. Since $E_{Sk}>\mathrm{\Delta }E$, skyrmions do not affect the activation gap $\mathrm{\Delta }{E}_{\mathrm{ph}}$. The values of $\mathrm{\Delta }E$ and $E_{Sk}$ in the regime $w_0/w_1<0.75$ (dashed line) are obtained by restricting the calculation to the spin-polarized sectors. While in this regime they do not correspond to low-energy excitations, we give them as an indication of the effect of $\tilde{F}$ on these quantities.

Figure 5

(a) Evolution of the many-body gap above the ferromagnetic FCI ground state upon adding a dispersion term of amplitude $W$ [Eq. (5)] for $w_0/w_1=0.8$. (b) Maximum value of the bandwidth to obtain a non-ero many-body gap ($W_c^{\mathrm{FCI}}$) or a fully polarized ground state ($W_c^{\mathrm{FM}}$).

Figure 6

PES of the spin-polarized ground state with $\tilde{N}=10$ holes in a Brillouin zone with $N_x\times N_y=5\times 6$ points for $w_0/w_1=0.8$, for a subsystem with $N_A=4$ and $N_A=5$. We use linearized coordinates for the total momentum $(K_{x,A},K_{y,A})$ in subsystem $A$. The black dashed line is guide to the eye, indicating the position of the entanglement gap. The total number of levels below the gap is 9975 for $N_A=4$ (335 for even $K_{x,A}$ and 330 for odd $K_{x,A}$) and 23 256 for $N_A=5$ (776 for $K_{y,A}=0$, 775 otherwise). This matches the expectation for the Laughlin $1/3$ state.

Figure 7

Evidence for a fractional Chern insulator ferromagnet at ${\nu }_{h,T}=3/5$. (a) Low-energy spectrum of the system with $N=18$ fermions and $N_s=30$ moiré sites, for $w_0/w_1=0.8$, in the sector $S_z=N/2$. (b) Finite-size extrapolation of the many-body gap $\mathrm{\Delta }E$ and ground state degeneracy splitting $\delta E$ for $w_0/w_1=0.8$.

Figure 8

(a) PES of the ${\nu }_{h,T}=3/5$ spin-polarized ground state manifold with $N=15$ and $N_s=25$, after particle-hole conjugation in the spin-polarized Hilbert space, for $w_0/w_1=0.8$, and a subsystem with $N_A=3$. We use linearized coordinates for the total momentum $(K_{x,A},K_{y,A})$ in subsystem $A$. (b) PES of the $\nu =2/5$ FQH fivefold degenerate ground state for $N=10$, $N_{\varphi }=25$ flux quanta, and $N_A=3$. The number of states below the black dashed line is the same in both systems.