# Fileset

[2024A00243G_FQAH accepted manuscript.pdf](https://mdr.nims.go.jp/filesets/3b8a0315-5c17-48d0-a253-6493c66cb0b9/download)

## Creator

Zhengguang Lu, Tonghang Han, Yuxuan Yao, Aidan P. Reddy, Jixiang Yang, Junseok Seo, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Liang Fu, Long Ju

## Rights

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s <a href="https://www.springernature.com/gp/open-science/policies/accepted-manuscript-terms">AM terms of use</a>, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1038/s41586-023-07010-7[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Fractional quantum anomalous Hall effect in multilayer graphene](https://mdr.nims.go.jp/datasets/454f96de-185c-45e2-ba5d-0716be530384)

## Fulltext

Fractional Quantum Anomalous Hall Effect in Multilayer Graphene  Zhengguang Lu1†, Tonghang Han1†, Yuxuan Yao1†, Aidan P. Reddy1, Jixiang Yang1, Junseok Seo1, Kenji Watanabe2, Takashi Taniguchi3, Liang Fu1, Long Ju1* 1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA.  2Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan 3Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan *Corresponding author. Email: longju@mit.edu †These authors contributed equally to this work.  The fractional quantum anomalous Hall effect (FQAHE), the analog of the fractional quantum Hall effect1 at zero magnetic field, is predicted to exist in topological flat bands under spontaneous time-reversal-symmetry breaking2–6. The demonstration of FQAHE could lead to non-Abelian anyons which form the basis of topological quantum computation7–9. So far, FQAHE has been observed only in twisted MoTe2 at moiré filling factor v > 1/210–13. Graphene-based moiré superlattices are believed to host FQAHE with the potential advantage of superior material quality and higher electron mobility. Here we report the observation of integer and fractional QAH effects in a rhombohedral pentalayer graphene/hBN moiré superlattice. At zero magnetic field, we observed plateaus of quantized Hall resistance Rxy = ℎ𝑣𝑒2 at v = 1, 2/3, 3/5, 4/7, 4/9, 3/7 and 2/5 of the moiré superlattice respectively, accompanied by clear dips in the longitudinal resistance Rxx. Rxy equals 𝟐𝒉𝒆𝟐  at v =  and varies linearly with v — similar to the composite Fermi liquid (CFL) in the half-filled lowest Landau level at high magnetic fields14–16. By tuning the gate displacement field D and v, we observed phase transitions from CFL and FQAH states to other correlated electron states. Our system provides an ideal platform for exploring charge fractionalization and (non-Abelian) anyonic braiding at zero magnetic field7–9,17–19, especially considering a lateral junction between FQAHE and superconducting regions in the same device20–22.   The fractional quantum Hall effect observed in conventional two-dimensional electron gas (2DEGs) at the semiconductor interface is a classic example of intertwined electron correlation and topology effects in mailto:longju@mit.educondensed matter physics1. It has been proposed that similar exotic states could exist at zero magnetic field, by engineering flat electronic bands with non-zero Chern numbers 2–6. As a precursor of FQAHE, the integer quantum anomalous Hall effect (IQAHE) has been conceived23 and realized in magnetic topologic insulators and the moiré superlattice made of two-dimensional materials24–27. Recently, FQAHE has been proposed in twisted transition metal dichalcogenides (TMD) moiré superlattices28–32 and observed in twisted MoTe2 (t-MoTe2) at filling factors v > 1/210–13. Graphene-based moiré superlattices have also been proposed to host FQAH states33–38, and could potentially show a plethora of fractional states due to the fewer defects in graphene lattice than in TMD lattices. Although fractional Chern insulators have been observed in graphene-based moiré superlattices at high magnetic fields39,40, observation of FQAHE in any graphene system has not been reported to date. The moiré superlattice formed between rhombohedral graphene (RG) and hexagonal boron nitride (hBN) has been demonstrated to be a remarkable platform for emergent quantum phenomena. In the highly tunable electronic bands in crystalline multilayer RG, correlated insulators, superconductivity, Chern insulators, orbital magnetism and multiferroicity have been demonstrated (see Methods). When placed on hBN to form a moiré superlattice, RG exhibits Mott insulators, tunable ferromagnetism and Chern insulators, as well as superconductivity (see Methods). Based on a tight-binding calculation, the band dispersion at zero gate-displacement field D becomes flatter as the layer number increases from N = 2 to 5, while the valence band starts to suffer from increased trigonal warping with N > 541. At non-zero Ds, states near the band edge acquire larger Berry curvatures41,42 that enable the formation of topological flat band when a moiré potential from hBN is introduced36,43. Therefore, engineering topological flat bands in gate-tuned multilayer RG/hBN is a promising approach for realizing FQAHE. From an experimental point of view, the RG/hBN moiré superlattice features three advantages over twisted TMD moiré superlattices: 1. the higher material quality of graphene over TMD; 2. better electrical contact to graphene than to TMD; 3. the twist-angle inhomogeneity induces less variations of the moiré period in a hetero-bilayer moiré than in a homo-bilayer moiré. While the Chern insulator state has been observed in trilayer RG/hBN at integer moiré fillings (see Methods), FQAHE has not been found in this system so far.  Here we report observations of IQAHE and FQAHEs in a new graphene moiré system formed by pentalayer RG and hBN, as shown in Fig. 1a. While the general possibility of FQAHE in RG/hBN moiré superlattices has been suggested before36, the pentalayer system we study has not received any concrete theoretical analysis or predictions. The superlattice period is ~11.5 nm and the twist angle is ~0.77°. We observed quantized Hall resistance Rxy = ±ℎ𝑒2 at a moiré superlattice filling factor v = 1 and zero magnetic field. These states correspond to Chern numbers C = ±1. At fractional filling factors between 0 and 1, we observed 6 states with fractionally quantized Hall resistances Rxy = ℎ𝜈𝑒2 at zero magnetic field. We observed Rxy = 2ℎ𝑒2 at v = 1/2 and a linear dependence of Rxy on v. By tuning D, we observed phase transitions from composite Fermi liquid (CFL) to valley-polarized Fermi liquid and a correlated insulator. We have measured two devices with similar moiré superlattice periods and they both show IQAHE and FQAHE. The data presented in the main text is based on Device 1, while the data from Device 2 is included in Extended Data Figures 8&9. Phase Diagram of the Moiré Superlattice Figure 1b&c show the longitudinal resistance Rxx and transverse resistance Rxy as functions of charge density ne (filling factor v) and D measured at mixing chamber temperature T = 10 mK. Rxx and Rxy have been symmetrized and anti-symmetrized using data collected at B = ±0.1 T (see Methods and Extended Data Figure 4). While most regions on the maps show small Rxx and Rxy, large values of resistances emerge in a tilted stripe region. At small v up to 1/2, large Rxx emerges, while Rxy shows large fluctuations around zero. At v = 2/5 to 1, large Rxy emerges and gradually changes with v. At v = 1, 2/3, 3/5, and 2/5, Rxx shows local minimums while Rxy shows plateaus as a function of v. As we show in the following sections, these states feature quantized Rxy as expected for IQAHE and FQAHEs. We note that the charge density corresponding to v = 1 in our system is about 5 times smaller than that in t-MoTe210–13, due to the larger moiré superlattice period.  The insulating state with large Rxx we observed at v ≤ 1/2 occupies a large continuous range of filling factor, which is distinct from correlated insulators at integer filling factors or generalized Wigner crystals at discrete fractional filling factors of moiré superlattices44,45. It is possible, for example, that electron crystallization happens within the flat moiré conduction band and leads to a Wigner crystal state at zero magnetic field46. At D < 0, we observed correlated insulating states at integer filling factors v = 2, 3 and 4 (see Extended Data Figure 1). The existence of topological states and correlated insulating states at opposite Ds are consistent with previous experiments on RG/hBN moiré superlattices (see Methods). In this work, we focus on the regime of large positive displacement fields. In the tilted stripe region, the large anomalous Hall signals indicates spontaneous valley polarization that breaks the time-reversal symmetry. As the D required to observe the anomalous Hall signal is quite high, at finite charge density, the re-distribution of charges is expected to partially screen the externally applied D. This picture may explain the slope of the striped region, in which a larger D is needed to maintain a narrow bandwidth as v increases. The observation of IQAHE and FQAHEs at v <= 1 further suggests the presence of a topological flat band. Fig 1d shows a moiré band structure calculated from a tight-binding model for pentalayer RG43, with an added phenomenological superlattice potential to account for moiré effects from the nearly aligned hBN layer. For a top-to-bottom interlayer potential difference  = 75 meV, this model produces a |C|=1 lowest conduction moiré band that is extremely narrow (bandwidth < 5 meV) and isolated from other bands with a global band gap. In contrast, the lowest valence moiré band is significantly broader (bandwidth > 45 meV), topologically trivial (C = 0), and energetically overlapping with adjacent moiré bands. Integer Quantum Anomalous Hall Effect Figure 2 shows detailed characterizations of the anomalous Hall state at v = 1. At D/0 = 0.97 V/nm, both Rxy and Rxx exhibit hysteretic behaviors under scanned magnetic field, as shown in Fig. 2a&b. At T = 0.1 K, Rxy is quantized at ℎ𝑒2  at zero magnetic field, while Rxx is smaller than 5 Ω. Rxy remains quantized up to at least 1.6 K and Rxx remains small in the same temperature range. Figure 2c&d show the B-dependence of Rxy and Rxx in the vicinity of v = 1 at D/0 = 0.97 V/nm. The anomalous Hall state persists to B = 0 T and exhibits a wide plateau in both Rxy and Rxx. The dispersion of this state agrees well with a Chern number C = ±1 state (indicated by the dashed line) according to the Streda’s formula. Starting from ~0.6 T, more dips in Rxx emerge and their slopes with B agree with C = 2 and 3 states (indicated by additional dashed lines). Figure 2e shows the ne (v) dependence of Rxy and Rxx and features a plateau of width ne ~ 3*1010 cm-2. The quantized Rxy and small Rxx values of the state at v = 1 exist in a wide range of D, as shown in Fig. 2f. At both higher and lower Ds, the device shows small Rxx and Rxy, except for a peak of Rxx during both transitions. These observations indicate the IQAHE with C = ±1 at filling factor v = 1. The features corresponding to C = 2 and 3 are due to integer quantum Hall effects that emerge at a low magnetic field, which demonstrates the high electron mobility of our device. The incremental change C = 1 between these three features indicates that the isospin degeneracy is completely lifted at v = 1, which corresponds to one electron per moiré unit cell. The width of the IQAHE plateau corresponds to a ~10 times smaller charge density than that in t-MoTe211,13.  Fractional Quantum Anomalous Hall Effect  As shown by Fig. 1b&c, large anomalous Hall response is found in a wide range between v = 2/5 and 1. Figure 3a&b show finer maps of Rxx and Rxy in this range, where additional vertical line features can be seen. To better visualize the states corresponding to these lines, we present in Fig. 3c line-cuts along the dashed lines in Fig. 3a&b. We can observe plateaus of Rxy at v = 2/5, 3/7, 4/9, 4/7, 3/5 and 2/3 with the value of Rxy quantized at ℎ𝜈𝑒2. At the same time, Rxx shows clear dips at these filling factors, similar to the observations of fractional quantum Hall states in 2DEGs at high magnetic fields1,14. We note that the v = 2/5 state is right next to the boundary of the anomalous Hall region, which shows a peak in Rxx. Nevertheless, the clear plateau of Rxy and dip in Rxx are observed at v = 2/5. Figure 3d-f & h-j show the magnetic hysteresis scans at fractional filling factors corresponding to the states identified in Fig. 3c. For all these states, Rxy shows quantized values of ℎ𝜈𝑒2 while Rxx is much smaller than Rxy. Lastly, Fig. 3g&k shows the Landau-fan diagram of Rxx in the range of v < 1/2 and v > 1/2, respectively. The dips at fractional filling factors evolve into tilted lines whose slopes agree well with the dashed lines, which are calculated based on the Streda’s formula 𝜕𝑛𝑒𝜕𝐵=𝜈𝑒ℎ for the corresponding filling factors. As a function of D, all FQAH states develops a plateau of Rxy at the corresponding quantized values (see Extended Data Figure 2) and dips of Rxx. The center of the Rxy plateau and the dips in Rxx shift to higher D as the filling factor increases, which agrees with the tilted stripe shape of the anomalous Hall region as shown in Fig. 1b&c. The observations of quantized Rxy and the corresponding dips in Rxx at fractional filling factors, together with the hysteresis enclosing zero magnetic field indicate FQAH states in our graphene-based moiré superlattice. These states resemble the Jain sequence of fractional quantum Hall states14–16, but at zero magnetic field. Compared with t-MoTe2 where FQAHEs are only observed at v > 1/210–13, the fractional states we observed reside at both sides of the half-filling. This is likely due to the better electrical contact in graphene devices than that has been achieved in semiconductor devices at low charge densities. The narrowest plateau width of FQAH states we have observed is ~1010 cm-2, which is about 10 times narrower than the 3/5 state observed in t-MoTe211.  AHE and Phase Transitions at v = 1/2 In addition to FQAHE states that have plateaus in Rxy and dips in Rxx, we observed a continuously changing anomalous Hall resistance in a wide range of filling factors from 4/9 to 4/7. Especially at around v = 1/2, Rxy varies roughly linearly with ne and v while Rxx does not show any clear dips, as shown in Fig. 4a. At v = 1/2, Rxy equals 2ℎ𝑒2  . Fig. 4b shows the hysteretic behaviors of Rxy and Rxx under a sweeping magnetic field. Figure 4c shows the D-dependence of Rxy and Rxx at fixed filling factor v = 1/2. The value of Rxy spans in a plateau which is concurrent with small values of Rxx. At higher D, Rxy decreases monotonically while Rxx first increases and then decreases. At lower D, Rxx shoots up and Rxy shows large fluctuations in the range of 0.85 to 0.9 V/nm. At even lower D, Rxy becomes almost zero while Rxx remains at a few k. At higher D than that of the Rxy = 2ℎ𝑒2 plateau, Rxy decreases monotonically while Rxx first increases and then decreases. As shown in Fig. 4d, the state at D/0 = 0.97 V/nm features small values of Rxy and Rxx, but a decent value of the Hall angle 𝜃𝐻~9.5° corresponding to tan𝜃𝐻 = 𝑅𝑥𝑦𝑅𝑥𝑥 ~0.17. The absence of a dip in Rxx at v = 1/2 and the linear dependence of Rxy with v in the neighborhood are distinct from FQAH states we described in the previous section and suggest the absence of a charge gap11,47,48. These latter properties are reminiscent of the CFL in the half-filled lowest Landau level of 2DEGs at high magnetic fields14–16. Starting from the zero-magnetic-field CFL state, our data suggests two distinct types of phase transitions driven by D. At the higher D side of the CFL state, the system is in a valley-polarized metallic state, due to the non-zero anomalous Hall resistance and small Rxx. The persistence of valley polarization and the peak of Rxx at intermediate Ds suggest that it is a continuous phase transition from CFL to Fermi liquid (FL). This new type of phase transition has been recently proposed by theory in FQAHE systems but was not observed in t-MoTe249,50. At the lower D side, we have a phase transition from CFL to our observed correlated insulator at v ≤ 1/2. Our observations call for further experiments to explore both types of phase transition, which are beyond the scope of this work. Conclusion and Outlook We observed IQHAE at v = 1 and FQAHEs at both sides of half-filling of the first moiré conduction band.  Beyond the specific moiré superlattice demonstrated here, our results indicate the great potential of similar RG/hBN systems with varied layer number, gate displacement field and twist angle for FQAHE studies—an opportunity that has been largely overlooked by theory and experiment so far. Given the high material quality, additional opportunities of researching novel quantum phase transitions, electron crystals at zero magnetic field, and behaviors of CFL in the moiré potential are within the reach of experiments47,48. The possibility of high-Chern-number flat bands in RG (see Methods) also points to possibly more exotic FQAH states with non-Abelian anyons for topological quantum computation7–9. Furthermore, the co-existence of FQAHE and superconductivity (see Methods) in graphene systems facilitates the realization of synthetic non-Abelian anyonic braiding by using a lateral junction within the same device20–22.  References 1. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys Rev Lett 48, 1559–1562 (1982). 2. Neupert, T., Santos, L., Chamon, C. & Mudry, C. Fractional quantum Hall states at zero magnetic field. Phys. Rev. Lett. 106, 236804 (2011). 3. Tang, E., Mei, J. W. & Wen, X. G. High-temperature fractional quantum Hall states. Phys. Rev. Lett. 106, 236802 (2011). 4. N Regnault, B. B. Fractional Chern insulator. Phys. Rev. X 1, 021014 (2011). 5. Sheng, D. N., Gu, Z. C., Sun, K. & Sheng, L. Fractional quantum Hall effect in the absence of Landau levels. Nat. Commun. 2, (2011). 6. Sun, K., Gu, Z., Katsura, H. & Das Sarma, S. Nearly flatbands with nontrivial topology. Phys Rev Lett 106, 236803 (2011). 7. Moore, G. & Read, N. Nonabelions in the fractional quantum hall effect. Nucl Phys B 360, 362–396 (1991). 8. Wen, X. G. Non-Abelian statistics in the fractional quantum Hall states. Phys Rev Lett 66, 802 (1991). 9. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev Mod Phys 80, 1083–1159 (2008). 10. Cai, J. et al. Signatures of Fractional Quantum Anomalous Hall States in Twisted MoTe2. Nature 2023 1–3 (2023) doi:10.1038/s41586-023-06289-w. 11. Park, H. et al. Observation of Fractionally Quantized Anomalous Hall Effect. Nature (2023) doi:10.1038/S41586-023-06536-0. 12. Zeng, Y. et al. Thermodynamic evidence of fractional Chern insulator in moiré MoTe2. Nature (2023) doi:10.1038/S41586-023-06452-3. 13. Xu, F. et al. Observation of Integer and Fractional Quantum Anomalous Hall Effects in Twisted Bilayer MoTe2. Phys Rev X 13, 031037 (2023). 14. Willett, R. et al. Observation of an even-denominator quantum number in the fractional quantum Hall effect. Phys Rev Lett 59, 1776 (1987). 15. Jain, J. K. Composite-fermion approach for the fractional quantum Hall effect. Phys Rev Lett 63, 199 (1989). 16. Halperin, B. I., Lee, P. A. & Read, N. Theory of the half-filled Landau level. Phys Rev B 47, 7312 (1993). 17. Nakamura, J., Liang, S., Gardner, G. C. & Manfra, M. J. Direct observation of anyonic braiding statistics. Nature Physics 2020 16:9 16, 931–936 (2020). 18. Bartolomei, H. et al. Fractional statistics in anyon collisions. Science 368, 173–177 (2020). 19. Kundu, H. K., Biswas, S., Ofek, N., Umansky, V. & Heiblum, M. Anyonic interference and braiding phase in a Mach-Zehnder interferometer. Nat Phys 19, 515–521 (2023). 20. Lindner, N. H., Berg, E., Refael, G. & Stern, A. Fractionalizing majorana fermions: Non-abelian statistics on the edges of abelian quantum hall states. Phys Rev X 2, 041002 (2012). 21. Clarke, D. J., Alicea, J. & Shtengel, K. Exotic non-Abelian anyons from conventional fractional quantum Hall states. Nature Communications 2013 4:1 4, 1–9 (2013). 22. Vaezi, A. Fractional topological superconductor with fractionalized Majorana fermions. Phys Rev B Condens Matter Mater Phys 87, 035132 (2013). 23. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988). 24. Chang, C. Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013). 25. Deng, Y. et al. Quantum anomalous Hall effect in intrinsic magnetic topological insulator. Science 367, 895–900 (2020). 26. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020). 27. Li, T. et al. Quantum anomalous Hall effect from intertwined moiré bands. Nature 600, 641–646 (2021). 28. Wu, F., Lovorn, T., Tutuc, E., Martin, I. & Macdonald, A. H. Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers. Phys Rev Lett 122, 086402 (2019). 29. Li, H., Kumar, U., Sun, K. & Lin, S. Z. Spontaneous fractional Chern insulators in transition metal dichalcogenide moiré superlattices. Phys Rev Res 3, L032070 (2021). 30. Devakul, T., Crépel, V., Zhang, Y. & Fu, L. Magic in twisted transition metal dichalcogenide bilayers. Nature Communications 2021 12:1 12, 1–9 (2021). 31. Yu, H., Chen, M. & Yao, W. Giant magnetic field from moiré induced Berry phase in homobilayer semiconductors. Natl Sci Rev 7, 12–20 (2020). 32. Crépel, V. & Fu, L. Anomalous Hall metal and fractional Chern insulator in twisted transition metal dichalcogenides. Phys Rev B 107, L201109 (2023). 33. Ledwith, P. J., Tarnopolsky, G., Khalaf, E. & Vishwanath, A. Fractional Chern insulator states in twisted bilayer graphene: An analytical approach. Phys Rev Res 2, 023237 (2020). 34. Abouelkomsan, A., Liu, Z. & Bergholtz, E. J. Particle-Hole Duality, Emergent Fermi Liquids, and Fractional Chern Insulators in Moiré Flatbands. Phys Rev Lett 124, 106803 (2020). 35. Devakul, T. et al. Magic-angle helical trilayer graphene. Sci Adv (2023). 36. Zhang, Y. H., Mao, D., Cao, Y., Jarillo-Herrero, P. & Senthil, T. Nearly flat Chern bands in moiré superlattices. Phys Rev B 99, 075127 (2019). 37. Gao, Q., Dong, J., Ledwith, P., Parker, D. & Khalaf, E. Untwisting Moiré Physics: Almost Ideal Bands and Fractional Chern Insulators in Periodically Strained Monolayer Graphene. Phys Rev Lett 131, 096401 (2023). 38. Repellin, C. & Senthil, T. Chern bands of twisted bilayer graphene: Fractional Chern insulators and spin phase transition. Phys Rev Res 2, (2020). 39. Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018). 40. Xie, Y. et al. Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature 2021 600:7889 600, 439–443 (2021). 41. Koshino, M. & McCann, E. Trigonal warping and Berry’s phase Nπ in ABC-stacked multilayer graphene. Phys. Rev. B 80, 165409 (2009). 42. Zhang, F., Jung, J., Fiete, G. A., Niu, Q. & MacDonald, A. H. Spontaneous quantum Hall states in chirally stacked few-layer graphene systems. Phys. Rev. Lett. 106, 156801 (2011). 43. Park, Y., Kim, Y., Chittari, B. L. & Jung, J. Topological flat bands in rhombohedral tetralayer and multilayer graphene on hexagonal boron nitride moire superlattices. arXiv: 2304.12874 (2023). 44. Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 2020 579:7799 579, 359–363 (2020). 45. Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 2020 587:7833 587, 214–218 (2020). 46. Smoleński, T. et al. Signatures of Wigner crystal of electrons in a monolayer semiconductor. Nature 2021 595:7865 595, 53–57 (2021). 47. Goldman, H., Reddy, A. P., Paul, N. & Fu, L. Zero-Field Composite Fermi Liquid in Twisted Semiconductor Bilayers. Phys Rev Lett 131, 136501 (2023). 48. Dong, J., Wang, J., Ledwith, P. J., Vishwanath, A. & Parker, D. E. Composite Fermi Liquid at Zero Magnetic Field in Twisted MoTe2. Phys Rev Lett 131, 136502 (2023). 49. Barkeshli, M. & McGreevy, J. Continuous transitions between composite Fermi liquid and Landau Fermi liquid: A route to fractionalized Mott insulators. Phys Rev B Condens Matter Mater Phys 86, 075136 (2012). 50. Song, X.-Y., Zhang, Y.-H. & Senthil, T. Phase transitions out of quantum Hall states in moir\’e TMD bilayers. arXiv: 2308.10903 (2023).         Main Figure Legends  Fig. 1. Device configuration, topological flat band and phase diagram of the rhombohedral pentalayer graphene/hBN moiré superlattice. a. Schematic of the device configuration, showing a moiré superlattice between the top layer of graphene and the top hBN, with a moiré period of 11.5 nm. b-c. Phase diagrams of the device revealed by symmetrized Rxx and anti-symmetrized Rxy at B = ± 0.1 T as functions of ne (v) and D. The temperature at the mixing chamber of dilution refrigerator is 10 mK. Large anomalous Hall signals emerge in a tilted stripe region centered at D/0 ~ 0.93 V/nm. Clear dips of Rxx can be seen at filling factors of the moiré superlattice v = 1, 2/3, 3/5 and 2/5 (indicated by the dashed lines and arrows), where Rxy shows plateaus of values. d. Calculated band structure of our moiré superlattice with interlayer potential  = 75 meV, showing a flat |C|=1 moiré conduction band and a dispersive C=0 moiré valence band.  Fig. 2. Integer quantum anomalous Hall effect. a & b. Magnetic hysteresis scans of Rxy and Rxx at v = 1 and D/0 = 0.97 V/nm and T = 0.1 - 4 K. Solid (dashed) lines correspond to scanning B from positive (negative) values to negative (positive) values. At 0.1 K, Rxy is quantized at ±he2  which corresponds to a Chern number C = ±1, while Rxx shows a value < 5 Ω at B = 0 mT. c-d. Landau-fan diagrams of Rxx and Rxy at D/0 = 0.97 V/nm. The IQAH state can be seen as a wide plateau in both maps, which disperses with magnetic field with a slope that agrees well with the dashed lines (corresponding to C = ±1, as determined by the Streda’s formula). At above ~0.6 T, features associated with integer quantum Hall states start to appear, corresponding to C = ±2 and 3 correspondingly as indicated by additional dashed lines. e. Symmetrized Rxx and anti-symmetrized Rxy (we present the positive values for convenience) as functions of v at T = 10 mK and D/0 = 0.97 V/nm, featuring quantized Rxy in a plateau of width ~3*1010 cm-2. Inset: zoomed-in plot to reveal the plateau of Rxx < 5 Ω. f. Rxx and Rxy as a function of D at v = 1, featuring a wide plateau at ~0.8-1.03 V/nm. Moving to both higher and lower Ds, the device transitions to metallic states with small Rxx and Rxy. A peak in Rxx appears during both transitions.  Fig. 3. Fractional quantum anomalous Hall effects. a & b. Zoomed-in diagrams of symmetrized Rxx and anti-symmetrized Rxy (we present the positive values for convenience) at B = ± 0.1 T as functions of v (ne) and D. Fine features that could not be identified in Fig. 1d&e can be seen in the vicinity of v = 1/2, especially in the Rxx diagram. Data is collected using a constant voltage bias measurement. c. Rxx and Rxy along the dashed lines in a & b, taken with a constant current measurement. Clear plateaus of Rxy at 5h2e2, 7h3e2, 9h4e2,7h4e2, 5h3e2 and 3h2e2 emerge at v = 2/5, 3/7, 4/9, 4/7, 3/5 and 2/3, as indicated by the dashed lines and arrows. Rxx shows clear dips at the corresponding filling factors. d-f & h-j. Magnetic hysteresis scans of Rxy and Rxx at v = 2/5, 3/7, 4/9, 4/7, 3/5 and 2/3, showing quantized values of Rxy = hνe2 and much smaller Rxx. g&k. Landau-fan diagram of Rxx at D/0 = 0.92 V/nm. The FQAH states can be seen as tilted line features, the slopes of which agree well with the dashed lines. The slopes of dashed lines correspond to C = 2/5, 3/7 and 4/9 in g, and 4/7, 3/5 and 2/3 in k using the Streda’s formula.  Fig. 4. Anomalous Hall effect and phase transitions at half-filling. a. Symmetrized Rxx and anti-symmetrized Rxy (we present the positive values for convenience) at B = ± 0.1 T and D/0 = 0.93 V/nm in the neighborhood of half-filling. Rxy shows a value of 2he2 at v = 1/2 and varies roughly linearly with the change of filling factor, while no dip in Rxx is observed. These observations resemble the signatures of composite Fermi liquid (CFL) in 2DEGs at high magnetic fields. b. Magnetic hysteresis scans of Rxy and Rxx at v = 1/2, showing Rxy plateaus at ±2he2   and much smaller Rxx. c. Rxy and Rxx at v = 1/2 as functions of D, showing the same Rxy value in a plateau spanning from 0.9 to 0.94 V/nm. At higher D, both Rxy and Rxx decrease to close to zero. At lower D, Rxx shoots up while Rxy shows large fluctuations at around zero. d. Magnetic hysteresis scans of Rxy and Rxx at v = 1/2 and D/0 = 0.97 V/nm, showing anomalous Hall signals and a Hall angle 𝜃𝐻~9.5°, corresponding to tan𝜃𝐻 =𝑅𝑥𝑦𝑅𝑥𝑥~0.17. This indicates a phase transition from CFL to valley-polarized metal at the higher D side. At the lower D side, the phase transition happens between CFL and a correlated insulating state.        Methods Device fabrication The pentalayer graphene and hBN flakes were prepared by mechanical exfoliation onto SiO2/Si substrates. The rhombohedral domains of pentalayer graphene were identified using near-field infrared microscopy51, confirmed with Raman spectroscopy and isolated by cutting with a Bruker atomic force microscope (AFM)52. The van der Waals heterostructure was made following a dry transfer procedure. We picked up the top hBN, graphite, middle hBN, and the pentalayer graphene using polypropylene carbonate (PPC) film and landed it on a prepared bottom stack consisting of an hBN and graphite bottom gate. The device was then etched into a Hall bar structure using standard e-beam lithography (EBL) and reactive-ion etching (RIE). We deposited Cr/Au for electrical connections to the source, drain and gate electrodes.  Previous experimental works on multilayer rhombohedral graphene Multilayer RG with layer number > 2 has been studied heavily in experiment. In the highly tunable electronic bands in crystalline multilayer RG53–56, correlated insulators, superconductivity, Chern insulators, orbital magnetism and multiferroicity have been demonstrated53,56–63. When placed on hBN to form a moiré superlattice, RG exhibits Mott insulators, tunable ferromagnetism and Chern insulators, as well as superconductivity64–67 Previous experiments adopted three different device configurations: 1. RG on SiO2 substrate53–55; 2. Suspended RG56,57; 3. RG-encapsulated by hBN58–67. Our devices are in the third category, which balance high device quality, large parameter space enabled by high gate voltages, and multiple terminals that allow the decoupling of Rxx and Rxy. These factors all contribute to the experimental observation of FQAHE. Transport measurement The device was measured in a Bluefors LD250 dilution refrigerator with an electronic temperature of around 100 mK. Stanford Research Systems SR830 lock-in amplifiers were used to measure the longitudinal and Hall resistance Rxx and Rxy with an AC frequency at 17.77 Hz. For constant voltage measurement, a voltage bias of 60 uV was applied. For constant current measurement, a current bias of 2 nA was applied. Keithley 2400 source-meters were used to apply top and bottom gate voltages. Top-gate voltage Vt and bottom-gate voltage Vb are swept to adjust doping density ne = (CtVt + CbVb)/e and displacement field D/0 = (CtVt - CbVb)/2, where Ct and Cb are top-gate and bottom-gate capacitance per area calculated from the Landau fan diagram.  Disentangling longitudinal and Hall resistance The optical images of the two devices are shown in Extended Data Figure 1c&d. Two contacts on one side of the device did not work and were floated during the measurement. The device geometry is such that no pair of contacts allows a perfect measurement of pure Rxy or Rxx. To disentangle Rxx and Rxy in the resistance tensor, we used magnetic field symmetrization/anti-symmetrization method26,59,68. The longitudinal resistance and Hall resistance can be separated by using magnetic field symmetrization/anti-symmetrization method based on the fact that Rxx is symmetric with B, while Rxy is antisymmetric. Measurements performed at opposite magnetic fields (larger than the coercive field) can thus be used to extract Rxx and Rxy at B = 0 for the QAH states, following:    Rxx(0) = (R(B) + R(-B))/2                                               Rxy(0) = (R(B) - R(-B))/2  In Extended Data Figure 4. we demonstrate the field symmetrization/anti-symmetrization method step by step from the raw data to the plot we used in Fig. 1b&c as an example. For simplicity, R12,34 corresponds to a resistance measured by applying current from contact #1 to #2 and measuring the voltage between #3 and #4. R13,24 is measured at B = ±100mT as shown in Extended Data Figure 4a&b. The Rxy can be obtained by extracting the field antisymmetric part of R13,24 or R24,13, following: Rxy (0) = (R13,24(100mT)-R13,24(-100mT))/2 Since the longitudinal component is symmetric respect to magnetic field, they will be cancelled by this subtraction process at the same magnitude of magnetic field but with opposite sign. Therefore, the disentangled Rxy could be obtained which is plotted in Extended Data Figure 4c and used in Fig. 1c. Following the same argument, resistance measurement of R23,14 is performed at B = ±100mT as shown in Extended Data Figure 4d&e. The longitudinal resistance can be separate by following: Rxx (0) = (R23,14(100mT) + R23,14(-100mT))/2 Since the Hall resistance is anti-symmetric respect to magnetic field, they will be perfectly removed by this summation process at opposite magnetic field. So that longitudinal resistance can be separated and plot in Extended Data Figure 4f and also shown in Fig. 1b.    In Fig. 1c and 2c we plotted (Rxy (0.1 T) - Rxy (-0.1 T))/2, while in 2e&f, 3b&c and 4a&c we plotted (Rxy (-0.1 T) - Rxy (0.1 T))/2 for the convenience of presentation. Apart from disentangling longitudinal and Hall resistance, measurement at a magnetic field larger than the coercive field also eliminates the random fluctuations of the magnetic domains at zero magnetic field, which is typical for micro-sized devices. As shown in Extended Data Figure 6a, spikes appear randomly in all curves at B = 0 as we scan the moiré filling factor v. These fluctuations correspond to the random switching of the orbital magnetization in our micron-sized devices. At B = ± 100mT, these fluctuations are completely suppressed. We have also performed scans at even smaller magnetic fields. As shown in Extended Data Figure 6b, all the curves at B = 10mT, 50mT and 100mT overlap well. Therefore, the magnetic field of ±0.1T we used is just to avoid fluctuations, while the phenomena we observed is robust at B = 0. Magnetic hysteresis measurements of Rxx and Rxy The four-probe magnetic hysteresis measurements with all possible pin combinations and the results of raw data and data after processing at v=3/5 in Extended Data Figure 5 as an example. First, we used magnetic field symmetrization/anti-symmetrization method to disentangle Rxx and Rxy. Magnetic field sweep of R13,24 is shown as solid (dashed) line in Extended Data Figure 5a. In this case, Hall resistance can be extracted by following the field anti-symmetrization method: Rxy_solid=(R13,24_solid(+B)-R13,24_dash(-B))/2.                   Rxy_dash=(R13,24_dash(+B)-R13,24_solid(-B))/2 The obtained Rxy is plotted in Extended Data Figure 5g with anomalous Hall value equals to the quantized value at v=3/5. Based on the raw data shown in Extended Data Figure 5b, magnetic field sweep of R24,13 can be treated in a similar way: Rxy_solid=(R24,13_solid(+B)-R24,13_dash(-B))/2.                   Rxy_dash=(R24,13_dash(+B)-R24,13_solid(-B))/2 The hysteresis loop after anti-symmetrization is shown in Extended Data Figure 5h. The magnetic sweep of longitudinal resistance can be extracted based on the raw data from R14,23 and R23,14 measurements by following the magnetic field symmetrization method: Rxx_solid=(R14,23_solid(+B)+R14,23_dash(-B))/2.                   Rxx_dash=(R14,23_dash(-B)+R14,23_solid(+B))/2 Rxx_solid=(R23,14_solid(+B)+R23,14_dash(-B))/2.                   Rxx_dash=(R23,14_dash(-B)+R23,14_solid(+B))/2 The symmetrized Rxx is plotted in Extended Data Figure 5i&l.  Other than relying on the symmetrization/anti-symmetrization with respect to magnetic field, we also performed symmetrization/anti-symmetrization by using Onsager reciprocal relation to better capture the behaviors of the magnetic domain26,59,68. R14,23 and its Onsager reciprocal R23,14 are mainly contributed by Rxx. The symmetrization can then be performed by following: Rxx_solid=(R14,23_solid+R23,14_solid)/2.                                Rxx_dash=(R14,23_dash+R23,14_dash)/2 The hysteresis loop of the disentangled longitudinal resistance obtained by Onsager reciprocal relation is plotted in Extended Data Figure 5n. Similarly, R24,13 and its Onsager reciprocal R13,24 are dominated by Rxy. The longitudinal part can be eliminated by performing the following anti-symmetrization process: Rxy_solid=(R13,24_solid−R24,13_solid)/2.                               Rxy_dash=(R24,13_dash−R13,24_dash)/2 As we can see from Extended Data Figure 5c,f&m, no matter what symmetrization method we use, the Hall resistance value always stay quantized at B = 0. Since the physics we would like to address in the manuscript is not about magnetic domain, symmetrization/anti-symmetrization respect to magnetic field is used for the hysteresis loop in Fig. 2a&b, Fig. 3d-j and Fig. 4b&d. Phenomenological model for moiré bands We calculate the band structure shown in Fig. 1d starting from the 𝑘 ⋅ 𝑝 Hamiltonian of Ref43 for (moiré-less) N-layer RG with N=5 and 𝑡0 = −2.6 eV for nearest-neighbor intralayer hopping. Using the hBN moiré potential model of Ref43, which only couples directly to the adjacent graphene layer, we do not obtain an isolated |C|=1 band at large displacement field. We note, however, that the nearly aligned hBN layer will induce a charge density modulation and the resulting Hartree-Fock potential will act on all layers. This motivates us to consider the following phenomenological model: 𝐻 = 𝐻0(𝑘⃗  ) + 𝑉(𝑟 ) where 𝐻0 (𝑘⃗ ) is the moiré-less  𝑘 ⋅ 𝑝 Hamiltonian and  𝑉(𝑟 ) = −2𝑉0 ∑ cos(𝑔 𝑖 ⋅ 𝑟 + 𝜙)𝑖=1,3,5 is a superlattice potential that acts identically on all layers. We choose 𝑉0 = 10 meV, 𝜙 = 0. We note that the appearance of an isolated |C|=1 band does not rely on fine tuning of these parameters or the potential acting equally on all layers. Here the moiré reciprocal lattice vectors are  𝑔 1 = (𝜖 − 𝜃 𝑧̂ ×)(0,4𝜋𝑎0√3) where 𝜖 = 0.017 is the graphene-hBN lattice constant mismatch and we choose 𝜃 = 0.744𝜋/180 to achieve a moiré period 𝑎𝑀 =4𝜋√3|𝑔⃗ 1|= 11.5 nm as determined experimentally.  Methods only references 51.  Ju, L. et al. Topological valley transport at bilayer graphene domain walls. Nature 520, 650–655 (2015). 52. Li, H. et al. Electrode-Free Anodic Oxidation Nanolithography of Low-Dimensional Materials. Nano Lett 18, 8011–8015 (2018). 53. Bao, W. et al. Stacking-dependent band gap and quantum transport in trilayer graphene. Nat. Phys. 7, 948–952 (2011). 54. Zhang, L., Zhang, Y., Camacho, J., Khodas, M. & Zaliznyak, I. The experimental observation of quantum Hall effect of l=3 chiral quasiparticles in trilayer graphene. Nature Physics 2011 7:12 7, 953–957 (2011). 55. Zou, K., Zhang, F., Clapp, C., MacDonald, A. H. & Zhu, J. Transport studies of dual-gated ABC and ABA trilayer graphene: Band gap opening and band structure tuning in very large perpendicular electric fields. Nano Lett 13, 369–373 (2013). 56. Lee, Y. et al. Competition between spontaneous symmetry breaking and single-particle gaps in trilayer graphene. Nat. Commun. 5, (2014). 57. Myhro, K. et al. Large tunable intrinsic gap in rhombohedral-stacked tetralayer graphene at half filling. 2d Mater 5, 045013 (2018). 58. Shi, Y. et al. Electronic phase separation in multilayer rhombohedral graphite. Nature 2020 584:7820 584, 210–214 (2020). 59. Zhou, H. et al. Half- and quarter-metals in rhombohedral trilayer graphene. Nature 2021 598:7881 598, 429–433 (2021). 60. Zhou, H., Xie, T., Taniguchi, T., Watanabe, K. & Young, A. F. Superconductivity in rhombohedral trilayer graphene. Nature 2021 598:7881 598, 434–438 (2021). 61. Han, T. et al. Correlated Insulator and Chern Insulators in Pentalayer Rhombohedral Stacked Graphene. Nat. Nanotechnology (2023). 62. Han, T. et al. Orbital Multiferroicity in Pentalayer Rhombohedral Graphene. Nature (2023). 63. Liu, K. et al. Interaction-driven spontaneous broken-symmetry insulator and metals in ABCA tetralayer graphene. Nat Nanotechnol (2023). 64. Chen, G. et al. Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice. Nature Physics 2019 15:3 15, 237–241 (2019). 65. Chen, G. et al. Tunable Orbital Ferromagnetism at Noninteger Filling of a Moiré Superlattice. Nano Lett 22, 238–245 (2022). 66. Chen, G. et al. Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice. Nature 579, 56–61 (2020). 67. Chen, G. et al. Signatures of tunable superconductivity in a trilayer graphene moiré superlattice. Nature 572, 215–219 (2019). 68. Sample, H. H., Bruno, W. J., Sample, S. B. & Sichel, E. K. Reverse‐field reciprocity for conducting specimens in magnetic fields. J Appl Phys 61, 1079–1084 (1987).    Acknowledgments  We acknowledge helpful discussions with X.G. Wen, T. Senthil, P. Lee, F. Wang and R. Ashoori. We thank D. Laroche for assistance with early investigation of a related sample. L.J. acknowledges support from a Sloan Fellowship. Work by T.H., J.Y. and J.S. was supported by NSF grant no. DMR- 2225925. The device fabrication of this work was supported by the STC Center for Integrated Quantum Materials, NSF grant no. DMR-1231319 and was carried out at the Harvard Center for Nanoscale Systems and MIT.Nano. Part of the device fabrication was supported by USD(R&E) under contract no. FA8702-15-D-0001. K.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Numbers 20H00354, 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. A.R. was supported by the Air Force Office of Scientific Research (AFOSR) under award FA9550-22-1-0432. L.F. was supported by the STC Center for Integrated Quantum Materials (CIQM) under NSF award no. DMR-1231319.  Author Contributions L.J. supervised the project. Z.L. and T.H. performed the DC magneto-transport measurement. T.H. and Y.Y. fabricated the devices. J.Y., J.S., Z.L. and T.H. helped with installing and testing the dilution refrigerator. A.R. and L.F. performed the calculations. K.W. and T.T. grew hBN crystals. All authors discussed the results and wrote the paper. Competing Interests The authors declare no competing interests. Correspondence and requests for materials should be addressed to Long Ju. Reprints and permissions information is available at http://www.nature.com/reprints. Data availability The data shown in the main figures are available from https://doi.org/10.7910/DVN/T4QPNP. Other data that support the findings of this study are available from the corresponding authors upon reasonable request.  http://www.nature.com/reprintshttps://doi.org/10.7910/DVN/T4QPNP%20.     Extended Data Figure Legends  Extended Data Figure 1. Phase diagram and optical micrographs of our devices. a&b corresponds to the hole-doping and electron-doping sides, respectively. The hole side shows resistive states at filling factors v = -2 and –4, while the electron side shows correlated insulating states at v = 2, 3 and 4 at D < 0—opposite side of D at which we observed IQAHE and FQAHEs. c. Device 1 from which the data in the main text is taken. Scale bar: 3µm  d. Device 2, the data of which is included in Extended Data Fig. 8&9. Scale bar:3µm.  Extended Data Figure 2. Gate displacement field dependence of Rxx and Rxy at fractional filling factors for Device 1. Each FQAH state shows quantized Rxy in a range of D, while the center of this range for different states shifts with the filling factor. The D corresponding to the minimum of Rxx also shifts with the filling factor in the same direction.  Extended Data Figure 3. Temperature dependence of FQAH states. a-f. Temperature dependence of Rxy. All states still remain quantized at 400 mK. g-l. Temperature dependence of Rxx.  Extended Data Figure 4. Symmetrization/anti-symmetrization method to obtain Fig. 1b&c. a,b&d,e. Raw data of R13,24 and R23,14 measured as functions of displacement field and moiré filling factor v at B=±100mT. The insets show the measurement pin configurations. c&f. Rxy and Rxx obtained after the symmetrization/anti-symmetrization process.    Extended Data Figure 5. Symmetrization/anti-symmetrization method to obtain magnetic hysteresis data at v=3/5. a, b, d, e, g, h, j&k. Raw data of R13,24, R24,13, R14,23 and R23,14 measured as functions of magnetic field. The insets show the measurement pin configurations. c, f, i&l. Rxy and Rxx obtained after the symmetrization process. m&n. Rxy and Rxx extracted after the symmetrization/anti-symmetrization process using the Onsager reciprocal relation.  Extended Data Figure 6. Rxy line scans at small magnetic fields. a. Rxy line scan versus moiré filling factor v at D/ε0 = 0.9 V/nm. Curves with rainbow colors represent multiple scans at B = 0. Black curves show scans at B = ± 100 mT. B. Rxy line scans versus v at B= 10 mT, 50 mT, 100 mT.    Extended Data Figure 7. Rxx line scans at varying magnetic field. a&b. Rxx line scans with moiré filling factor v < 1/2 and v > 1/2, respectively. Dips at fractional filling factors shift with magnetic field as indicated by the dashed lines. Curves are equally shifted vertically for clarity.  Extended Data Figure 8. Data from Device 2. a&b. Phase diagrams of the device revealed by symmetrized Rxx and anti-symmetrized Rxy at B = ± 0.1 T as functions of charge density ne (filling factor v) and D. The temperature at the mixing chamber of dilution refrigerator is 10 mK. Clear dips of Rxx can be seen at filling factors of the moiré superlattice v = 1, 2/3 and 2/5 (indicated by the dashed lines and arrows), where Rxy shows plateaus of values.   Extended Data Figure 9. Magnetic hysteresis data from Device 2. a-c. Magnetic hysteresis measurements at v = 1, 2/3 and 2/5. Clear hysteresis and values of Rxy at hve2can been seen.