A.H.M.: My group has been working to improve understanding of the origin of the FQAHE in some transition metal dichalcogenide (TMD) homobilayers. Our approach3,4 is motivated primarily by the observation5 that in this class of moiré materials, the Hamiltonian contains a topological skyrmion lattice field that acts on the layer degree of freedom. When this field is treated in an adiabatic approximation, valid at long moiré periods, it replaces the layer degree of freedom by Berry phases that can be represented by a periodic magnetic field with one flux quantum in each unit cell and gives rise to residual terms associated with potential and zero-point kinetic energies. We associate the FQAHE with approximate cancellation of these residual terms, which we argue is common in TMD homobilayer moiré materials. Following this line of argument, we have predicted that the FQAHE will not occur when pressure is applied to reduce the separation between layers and increase the strength of interlayer tunnelling.

Our work connects with other threads of FQAHE research. It turns out that the Bloch bands of 2D electrons in a periodic magnetic field with no residual terms have ideal quantum geometry6 and therefore a flat band Hilbert space in which wavefunctions are analytic and have one zero per unit cell. Low-energy states are constructed by placing zeros in the dependence of the many-body wavefunction on one coordinate at the positions of the other particles. This property can explain why the chemical potential jumps at certain rational band fillings. As the number of zeros changes when an external magnetic field is applied, it also explains why the chemical potential jump occurs at a magnetic-field-dependent density. I personally feel that this must be the essence of the FQAHE in TMD homobilayers. Currently, my group is working to understand the relationship between this continuum model explanation and FQAHE states in lattice models that represent TMD homobilayers more approximately and the relationship between real-space Berry phases and the momentum-space description of band topology, which seem to have a more natural connection to the FQAHE states observed by the group of L.J. in graphene multilayers.

X.X.: The theoretical prediction of the FQAHE in a new phase of matter — a zero-field FCI — has long captivated researchers. Despite this interest, realizing this phenomenon in practical systems has remained challenging. Inspired by works from Allan MacDonald5 (and later Wang Yao) predicting twisted MoTe2 bilayers (tMoTe2) to be a topological insulator with flat bands, with my group we carried out a systematic exploration of this platform.

Our optimism soared upon the observation of spontaneous ferromagnetic order at both integer and fractional fillings of the first flat Chern band in tMoTe2 at a twist angle of 3–4°, a crucial component for the FQAHE7. This pivotal finding, supported by theoretical frameworks8,9, strongly suggested that twisted MoTe2 could be the magic system for realizing the FQAHE. Building on this result, we were able to obtain a ‘fan diagram’ by tracking the three ferromagnetic incompressible states (at fillings v = −1, −2/3 and −3/5) using a trion-sensing technique. The analysis through the Streda formula, which relates the Hall conductance to the derivative of the carrier density of the gapped states with respect to the magnetic field, revealed that these three states possess slopes characteristic of integer and fractional zero-field FCI states, providing evidence of the FQAHE10.

Despite these advances, direct evidence of the FQAHE hinges on transport measurements, which have been impeded by technological hurdles in fabricating good metal contacts to 2D semiconductors. We had been tackling this challenge even before the promising results mentioned earlier. Once we knew the FQAHE existed in tMoTe2, our confidence and excitement facilitated a swift resolution of these fabrication issues. This advance allowed us to observe the quantized Hall resistance with suppressed longitudinal resistance at zero magnetic field, confirming both integer (−1) and fractional (−2/3 and −3/5) quantized anomalous Hall effects11. In addition, a putative composite Fermi-liquid state was detected at half-filling of the flat Chern band. With the application of an out-of-plane electric field, we induced a phase transition from the Chern insulator states to either Mott or (putative) charge-ordered insulating states. These findings provide definitive evidence of the FQAHE and establish tMoTe2 as a prime candidate for investigating topological quantum phase transitions. Most recently, in collaboration with the group of Zhi-Xun Shen at Stanford, we visualized the edge conduction channels of these zero-field FCI states using exciton-resonant microwave impedance microscopy12. The direct observation of the bulk-edge correspondence, that is, of the insulating bulk and of conduction edges at both integer and fractional fillings, provides an important confirmation of the FQAHE observed in transport.

J.S. and K.F.M.: We have two recent works in this area, both on twisted bilayer MoTe2. In the first study13, we observed a time-reversal breaking and a gapped state at 2/3-filling of the moiré lattice by measuring the chemical potential using an optical readout technique we very recently developed14. We showed that the state gains 2/3 of an electron per magnetic flux quantum threaded through the sample, which is a telltale sign of an FCI according to the Streda formula. We further demonstrated that an electric field perpendicular to the sample plane can induce a topological phase transition from the FCI to a gapless state. These results open the door for studies of fractionalized electrons and their quantum phase transitions in the absence of magnetic fields.

Our second work reports a fractional QSH state in a small-twist-angle (2.1°) MoTe2 device15. Similar to FCIs, fractional QSH states are topologically ordered states that support long-range entanglement; unlike FCIs, the fractional QSH state here is protected by spin conservation and is an example of symmetry-enriched topological order. By combining four-terminal and two-terminal nonlocal measurements, we demonstrated the emergence of a fractional QSH state at a hole filling factor of 3 of the moiré lattice. The state displays zero anomalous Hall conductivity and supports helical edge states that carry an edge conductance of 3/2 of the conductance quantum. We also show the emergence of a single, double and triple QSH state at filling factors of 2, 4 and 6, respectively. The robust conductance quantization in all these QSH states is a result of spin conservation in moiré TMDs. These results have attracted much theoretical attention: many ground states have been proposed for the observed fractional QSH state, including both time-reversal-symmetric and time-reversal-breaking candidates, both Abelian and non-Abelian candidates, and a possible ground state displaying charge-spin separation. Further experimental investigations will be key to clarifying the nature of this state.

L.J.: With my group, we found that crystalline five-layer graphene sitting on top of hexagonal boron nitride (hBN) with a very small twist angle (0.77°) exhibits six different states showing the FQAHE16. Experimentally, we measured the Hall resistance and longitudinal resistance of two devices at 0.4 K and observed fractional quantized Hall resistance at filling factors v = 2/3, 3/5, 4/7, 4/9, 3/7 and 2/5 of the moiré superlattice. This dc transport method is, by definition, the only way to establish the presence of the FQAHE, and it is how the FQHE was established. This discovery is one of the several results we have published on crystalline five-layer graphene since last year17. What motivated us to investigate crystalline graphene in the rhombohedral stacking order are its highly tunable flat bands and Berry curvatures — the ingredients needed for correlated and topological electron phenomena, respectively. The FQAHE is one of the states that we were hoping to find and probably the most exciting one we have observed so far.

The discovery of the FQAHE was still surprising though: no theorist predicted that this material would host such states, and this system is, at first glance, very different from twisted MoTe2. In twisted MoTe2, the charges feel a strong moiré potential and an emergent magnetic field owing to the strong spin–orbit coupling: both are believed to be key ingredients for engineering the FQAHE in twisted MoTe2. However, at the charge density and gate electric field where we observed the FQAHE in five-layer graphene, neither of these two ingredients is present. As a result, 6 months after we shared the results on the arXiv, the underlying mechanism is still under debate18,19. This experimental result is very intriguing for three reasons. First, it might teach us something fundamental about what is important to engineer the FQAHE: it seems that there could be more ways than we believed to generate these states. Second, graphene samples display high material quality, ease of contact fabrication and low spatial inhomogeneity, which will facilitate future research in this direction. Finally, crystalline graphene multilayers are known to exhibit superconductivity, which could enable the engineering of more exotic quasiparticles, known as parafermions, when combined with the FQAHE.