Fractional Chern insulators in magic-angle twisted bilayer graphene

Abstract

Fractional Chern insulators (FCIs) are lattice analogues of fractional quantum Hall states that may provide a new avenue towards manipulating non-Abelian excitations. Early theoretical studies^1-7 have predicted their existence in systems with flat Chern bands and highlighted the critical role of a particular quantum geometry. However, FCI states have been observed only in Bernal-stacked bilayer graphene (BLG) aligned with hexagonal boron nitride (hBN)⁸, in which a very large magnetic field is responsible for the existence of the Chern bands, precluding the realization of FCIs at zero field. By contrast, magic-angle twisted BLG^9-12 supports flat Chern bands at zero magnetic field^13-17, and therefore offers a promising route towards stabilizing zero-field FCIs. Here we report the observation of eight FCI states at low magnetic field in magic-angle twisted BLG enabled by high-resolution local compressibility measurements. The first of these states emerge at 5 T, and their appearance is accompanied by the simultaneous disappearance of nearby topologically trivial charge density wave states. We demonstrate that, unlike the case of the BLG/hBN platform, the principal role of the weak magnetic field is merely to redistribute the Berry curvature of the native Chern bands and thereby realize a quantum geometry favourable for the emergence of FCIs. Our findings strongly suggest that FCIs may be realized at zero magnetic field and pave the way for the exploration and manipulation of anyonic excitations in flat moiré Chern bands.

Fig. 1. Incompressible states with fractional quantum numbers in MATBG.

a, Local inverse compressibility dµ/dn measured as a function of magnetic field B and electrons per moiré unit cell ν. b, Wannier diagram identifying the incompressible peaks present in a. Black lines correspond to ChIs and integer quantum Hall (IQH) states; green lines correspond to correlated insulators (CIs) emanating with nonzero integer s and t = 0; blue lines correspond to CDWs with integer t = 0 and fractional s; yellow lines correspond to SBCIs with nonzero integer t and fractional s; and orange lines correspond to FCIs with fractional t and fractional s. Grey shaded regions correspond to the gaps to the remote bands. Source data

Fig. 2. Density wave states at low magnetic field for 2.5 < ν < 4.

a, Local inverse compressibility dµ/dn between ν = 2.5 and 4 for B = 0 T to 3 T. b, Wannier diagram corresponding to the states observed in a coloured according to the classification used in Fig. 1b. c, Band energy (purple) and Berry curvature (yellow) along a path through the Γ point in the first mini-Brillouin zone. Zero momentum corresponds to the Γ point. d, Band structure in the case of unit-cell (UC) doubling resulting in a C = ±1 band accompanied by a new C = 0 band. e, Band structure in the case of unit-cell tripling resulting in a C = ±1 band accompanied by two new C = 0 bands. f–h, Band fillings in the case of unit-cell doubling (f) and unit-cell tripling (g, h) needed to produce the density wave states observed in a. Source data

Fig. 3. FCIs in a weak magnetic field.

a, Local inverse compressibility dµ/dn between ν = 3 and 4 for B = 3 T to 11 T. b, Wannier diagram corresponding to the states observed in a coloured according to the classification used in Fig. 1b. Light blue and orange lines denote the CDWs and FCIs, respectively. The grey shaded region marks the energy gap to the remote band. c, Depiction of band fillings that lead to the (t, s) = (2/3, 10/3) FCI observed in a, which corresponds to a ν_c = 1/3 FCI state from the final C = −1 band populated on electron-doping the (1, 3) ChI. d, Chemical potential steps Δµ associated with the CDW and FCI states observed in a obtained by integrating the inverse compressibility dµ/dn. The error bars reflect the standard deviation obtained from fitting to μ(n). e, Calculated average Berry curvature deviation $\sigma \left(F\right)$ from the continuum model as a function of w₀/w₁. The grey shaded region corresponds to w₀/w₁ = 0.65 to 0.75, the range in which the transition from FCI to CDW occurs. This w₀/w₁ range allows us to estimate the range of values ${\sigma }_c\left(F\right)$ = 1.4 to 2.2 below which the FCI is favourable. f, Calculated average Berry curvature deviation $\sigma \left(F\right)$ as a function of magnetic field for w₀/w₁ = 0.8. g, Phase diagram constructed as a function of w₀/w₁ and magnetic field in units of ${\phi }_0$. The white lines indicate the contours ${\sigma }_{\mathrm{c}}\left(F\right)=1.4$ and ${\sigma }_{\mathrm{c}}\left(F\right)=2.2$ that define the region where the phase boundary between the FCI and CDW ground states is expected. Source data

Fig. 4. Additional FCIs at higher magnetic field.

a–d, Measurements of dµ/dn ($\times$10⁻¹¹ mV cm⁻²) in various density ranges between 6 and 12 T showing additional FCIs and SBCIs. e–h, Wannier diagrams corresponding to the states observed in a–d coloured according to the classification used in Fig. 1b. Green, yellow and orange lines denote the correlated insulators, SBCIs and FCIs, respectively. Source data

Extended data Fig. 1. Additional CDW and SBCI states at higher magnetic field.

a–d, Measurements of dµ/dn ($\times$10⁻¹¹ mV cm⁻²) in various density ranges between 6.5 and 12 T showing additional CDWs and SBCIs. e–g, Schematic Wannier diagrams corresponding to the states observed in a–d colored according to the classification used in Fig. 1b. Light blue and yellow lines denote the CDWs and SBCIs, respectively.

Extended Data Fig. 2. Energy gaps of additional FCI and SBCI states.

a–d, Chemical potential steps Δµ of the FCI (yellow and green circles) and SBCI (light and dark blue circles) states shown in Fig. 4.

Extended Data Fig. 3. Fits to (t, s) for FCIs and SBCIs.

a–e, Incompressible peak locations (blue circles) associated with FCI and SBCI states. Black lines mark the results of linear fits. The fitted slope of nearby Chern insulators were used to convert the parameters to (t, s), the values of which are shown in the brackets with 95% confidence intervals.

Extended Data Fig. 4. Effect of twist angle inhomogeneity on FCIs.

a, Local twist angle variation over a distance of 1.6 µm. b, Compressibility measurements between 𝜈 =3 and 4 measured along the same trajectory as in a at 9 T. The blue and black arrows indicate the densities near which the (2/3, 10/3) and (1/3, 11/3) states occur, respectively. c, Compressibility measurements taken at three locations indicated by the dotted lines in a. The white, blue, black and green dotted lines mark the expected evolution of the incompressible peaks associated with (1, 3), (2/3, 10/3), (1/3, 11/3) and (1, 13/4).

Extended Data Fig. 5. Hofstadter spectrum.

a, Calculated spectrum of the narrow bands of MATBG at finite magnetic field. The bands are colored according to their Chern number at zero magnetic field. b, Wannier plot corresponding to the spectrum in a.