The early 1980s saw two breakthroughs in condensed matter physics. First, it was discovered that the resistance of a 2D gas of electrons becomes quantized in the presence of a perpendicular magnetic field — the quantum Hall effect (QHE). The quantum theory of electrons in a magnetic field started in the 1930s and already implied unusual physics including macroscopic quantization of kinetic energy, called Landau level formation. It was nearly 50 years later, thanks to advances in semiconductor material growth methods, that it became possible to measure the resistance of 2D electrons experimentally. Resistance quantization, which turned out to be related to electron number quantization, came as a complete surprise. Second, it was discovered that the Hall resistance could exhibit fractional quantization (FQHE) at certain partial Landau level occupancies at which its slowly moving orbit centers develop strong correlations. Theorists soon understood that this implied the presence of fractionally charged quasiparticles in the system.

The FQHE stands as the best understood experimental realization of emergence and fractionalization in condensed matter, a theoretically important concept that is relevant for understanding quantum spin liquids and other strongly correlated electronic states. The fractional quasiparticles of the FQHE can sometimes have non-Abelian quantum statistics, meaning that the many-electron quantum state is altered when two quasiparticles are rotated around each other so that they exchange locations — a property that provides a potential resource for topological quantum computation.

In recent years, advances in material fabrication techniques have enabled a new type of 2D matter — moiré materials — artificial crystals with bands of states that can have slow electrons and strong correlations, like Landau levels. In 2023, fractional charge states were observed in special classes of 2D moiré systems1,2,3,4,5 in which the band wavefunctions are providentially also similar to those in Landau levels.

Necessary conditions to achieve fractionalization

The QHE was eventually understood to be a consequence of the quantized degeneracy of the discrete electronic kinetic energy eigenstates (Landau levels) of 2D electrons in the presence of strong perpendicular magnetic fields, combined with disorder-induced localization. When all Landau levels are filled or empty, a 2D electron system behaves as an insulator in the sense that there is a gap in the electronic band structure in the bulk of the system. Because the Landau level degeneracy is one state for every quantum of magnetic flux passing through the 2D layer, the density of the insulating state changes continuously when the magnetic field changes, and this property implies that 2D quantum Hall insulators always have conducting edges that carry circulating currents in equilibrium.

Not long after the observation of the integer quantum Hall effect, it was discovered that insulating states could also occur at fractional Landau level filling factors, implying fractionally charged quasiparticles. When an electron is added or removed from a FQH insulator it splits into fractionally charged electrons and holes that can be widely separated and act as independent degrees of freedom. The FQHE relies on electron–electron interactions that lead to very special forms of complex correlations between electrons. Compared to the QHE, the FQHE typically requires stronger magnetic fields and/or weaker disorder in the material, so that the interactions between quasiparticles are stronger than their interactions with unintended random potentials.

Theoretical research motivated by the QHE and FQHE led to the understanding first that the integer quantum Hall effect could in principle occur without a magnetic field6 and then that the fractional effect could also occur7,8 in magnetic materials (those in which time-reversal symmetry is spontaneously broken) with properties that are ‘just right’. The ordinary (non-quantized) Hall effect produced by magnetism instead of magnetic fields is referred to as anomalous — and this descriptor has been adopted for the quantized case so that the two quantized effects are referred to as the QAHE and FQAHE (Fig. 1). In the integer case the QAHE is understood to be connected to non-trivial topology in an electronic band characterized by a topological invariant known as a Chern number — hence the two states are also sometimes referred to as Chern insulators (CIs) and fractional Chern insulators (FCIs). The Chern number is integer valued and linked to momentum space Berry phases; geometric phases acquired by electrons that execute closed paths in momentum space, and can only be nonzero when time-reversal symmetry is broken. The theoretical identification of Chern insulators as a special class of magnets spawned interest in topological classifications of electronic bands more generally — a topic that has been a major theme of condensed matter physics over recent decades.

Fig. 1: The quantum Hall effects.
figure 1

a, The integer QHE. The number of states in a Landau level is equal to the number of quanta of magnetic flux that pass through the 2D system. When the Landau level is full there is a kinetic energy gap between Landau levels and adding or removing a flux quantum creates a quasiparticle (blue) with integer charge ±e. The dependence of Landau level degeneracy on field implies that finite systems have chiral edge states (indicated by arrows) that carry current without dissipation. b, The fractional QHE occurs when interactions open energy gaps at a fractional (suggested by lighter shading) Landau level filling factor ν. In the fractional case adding or removing a flux quantum adds a quasiparticle(s) with total charge νe, which is a fraction of the electron charge. c, Moiré materials are artificial crystals associated with a periodic moiré pattern formed by overlapped 2D crystals (light grey background). The integer QAHE occurs when time-reversal symmetry is broken to yield band degeneracies that are immediately magnetic field dependent starting at zero magnetic field. The edge states then represent an unusual type of orbital ferromagnetism. d, When interactions open gaps at fractional band filling ν, adding a magnetic flux quantum generates a fractionally charged quasiparticle (blue). When electrons are added or removed from a fractional insulator, they split up into a collection of fractionally charged quasiparticles whose states can be changed simply by circling one quasiparticle (red) around another (blue).

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The conditions required to realize fractional quantum anomalous Hall states are more subtle and continue to be clarified. In addition to broken time-reversal symmetry and non-zero Chern numbers, the electronic band needs to resemble Landau levels in other ways. Firstly, the electron velocities need to be small enough to make the interaction energy dominate over the kinetic energy. Secondly, the momentum-space quantum geometry of the band wavefunctions needs to be equivalent to that of Landau level — in which case it is referred to as ideal. In particular, the absolute value of the Berry curvature (the Berry phase flux) should be close to the trace of the quantum metric tensor — the analogy of the metric tensor of curved space. Thirdly, the Berry curvature should not be too inhomogeneous. Failure to satisfy these conditions typically results in other less exotic insulating states, for example Wigner crystals or some other type of charge density wave state.

The moiré materials platform

When 2D materials are isolated from van der Waals compounds and overlaid, they often have long-period spatial beating patterns; moirés, with periods inversely related to a difference in lattice constant or orientation between layers. When the host materials are 2D semiconductors or semimetals, like graphene or semiconductor transition metal dichalcogenides, electronic properties are accurately described by models that have the periodicity of the moiré pattern. These moiré materials, which can now be fabricated routinely, are effectively artificial 2D crystals with large real space unit cells and corresponding small (mini) momentum space unit cells. Crucially, their long periods allow the number of electrons per effective atom to be tuned with electrical gates over ranges that exceed one — allowing experimenters to roam freely across the moiré material periodic table9.

In TMD and graphene-based moiré materials, the electron bands form separately in the two distinct corners of the microscopic Brillouin zone referred to as valleys. Time-reversal symmetry can be broken by differential occupation of the moiré bands of the two valleys — a kind of magnetism that is decidedly unusual because time-reversal is broken directly in orbital instead of spin degrees of freedom. It turns out that the valley projected bands often have non-zero Chern invariants, explaining the anomalous integer QHE. The fractional case is driven by interactions and is therefore sensitive not only to momentum space topology but also to real space properties of Bloch state wavefunctions that are captured by the band’s quantum geometry.

Non-trivial quantum geometry has long been recognized in graphene and graphene multilayer systems, where it is related to momentum-dependent pseudospins provided by the two sublattices of the graphene honeycombs. It is more of a surprise in TMDs which have a single isolated band in each valley with a fixed atomic character. It turns out, however, that because the layers are identical and occupied simultaneously, TMD AA homobilayer moirés have an active layer degree of freedom with spatial structure that is just right10 to produce narrow Chern bands with nearly ideal quantum geometry. The effective magnetic field underlying the ideal quantum geometry can be traced to real space Berry phases produced by the position dependence of the layer pseudospin.

Recent findings

In 2023, work by several different groups1,2,3,4 has accumulated convincing experimental evidence for the FQAHE in MoTe2 homobilayer moiré superlattices. The FQAHE has now been observed in this moiré system in three different laboratories by measuring the Hall resistivity directly and also by optically detecting the magnetic field dependence of the density at which the 2D electron system is an insulator. More recent experiments5 have established using transport that a certain type of graphene multilayer stack, pentalayer rhomobohedral graphene, that is aligned with hexagonal boron nitride to form a moiré, also has a FQAHE.

Other transition metal dichalcogenide homobilayers (and heterobilayers, if they are dual-gated) may also yield a FQAHE, and this is yet to be explored. Much work remains to sort out the similarities and differences between the material systems, and to sort out the dependence on controllable moiré parameters like twist angle, and on temperature. Looking further ahead, moiré materials are a platform that allows localized quasiparticles to be directly accessed by scanning probes. If non-Abelian fractionalized states are discovered, this property could enable quantum computation by quasiparticle manipulation.