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Unlike most electronic topological phenomena, the fractional quantum Hall effect requires correlations among electrons. Spanton et al. describe a class of related but even more unusual states, the fractional Chern insulators (see the Perspective by Repellin and Regnault). They observed these states in samples of bilayer graphene, where one of the graphene layers was misaligned by a small angle with respect to an adjoining layer of hexagonal boron nitride. The misalignment created a superlattice potential and topologically nontrivial bands, which had a fractional filling, thanks to strong electronic interactions. The findings expand the class of correlated topological states, which have been predicted to harbor exotic excitations. Science , this issue p. 62 ; see also p. 31 Capacitance measurements on bilayer graphene together with calculations reveal a class of correlated topological states. Topologically ordered phases are characterized by long-range quantum entanglement and fractional statistics rather than by symmetry breaking. First observed in a fractionally filled continuum Landau level, topological order has since been proposed to arise more generally at fractional fillings of topologically nontrivial Chern bands. Here we report the observation of gapped states at fractional fillings of Harper-Hofstadter bands arising from the interplay of a magnetic field and a superlattice potential in a bilayer graphene–hexagonal boron nitride heterostructure. We observed phases at fractional filling of bands with Chern indices C = − 1 ,   ± 2 ,  and  ± 3 . Some of these phases, in C = − 1 and C = 2 bands, are characterized by fractional Hall conductance—that is, they are known as fractional Chern insulators and constitute an example of topological order beyond Landau levels.

Semiconductor nanowires with superconducting leads are considered promising for quantum computation. The current–phase relation is systematically explored in gate-tunable InAs Josephson junctions, and is shown to provide a clean handle for characterizing the transport properties of these structures. Gate-tunable semiconductor nanowires with superconducting leads have great potential for quantum computation 1 , 2 , 3 and as model systems for mesoscopic Josephson junctions 4 , 5 . The supercurrent, I , versus the phase, φ , across the junction is called the current–phase relation (CPR). It can reveal not only the amplitude of the critical current, but also the number of modes and their transmission. We measured the CPR of many individual InAs nanowire Josephson junctions, one junction at a time. Both the amplitude and shape of the CPR varied between junctions, with small critical currents and skewed CPRs indicating few-mode junctions with high transmissions. In a gate-tunable junction, we found that the CPR varied with gate voltage: near the onset of supercurrent, we observed behaviour consistent with resonant tunnelling through a single, highly transmitting mode. The gate dependence is consistent with modelled subband structure that includes an effective tunnelling barrier due to an abrupt change in the Fermi level at the boundary of the gate-tuned region. These measurements of skewed, tunable, few-mode CPRs are promising both for applications that require anharmonic junctions 6 , 7 and for Majorana readout proposals 8 .

We present transport and scanning SQUID measurements on InAs/GaSb double quantum wells, a system predicted to be a two-dimensional topological insulator. Top and back gates allow independent control of density and band offset, allowing tuning from the trivial to the topological regime. In the trivial regime, bulk conductivity is quenched but transport persists along the edges, superficially resembling the predicted helical edge-channels in the topological regime. We characterize edge conduction in the trivial regime in a wide variety of sample geometries and measurement configurations, as a function of temperature, magnetic field, and edge length. Despite similarities to studies claiming measurements of helical edge channels, our characterization points to a non-topological origin for these observations.

Ferromagnetism is most common in transition metal compounds where electrons occupy highly localized d orbitals. However, ferromagnetic order may also arise in low-density two-dimensional electron systems 1 – 5 . Here we show that gate-tuned van Hove singularities in rhombohedral trilayer graphene 6 drive spontaneous ferromagnetic polarization of the electron system into one or more spin and valley flavours. Using capacitance and transport measurements, we observe a cascade of transitions tuned to the density and electronic displacement field between phases in which quantum oscillations have fourfold, twofold or onefold degeneracy, associated with a spin- and valley-degenerate normal metal, spin-polarized ‘half-metal’, and spin- and valley-polarized ‘quarter-metal’, respectively. For electron doping, the salient features of the data are well captured by a phenomenological Stoner model 7 that includes valley-anisotropic interactions. For hole filling, we observe a richer phase diagram featuring a delicate interplay of broken symmetries and transitions in the Fermi surface topology. Finally, we introduce a moiré superlattice using a rotationally aligned hexagonal boron nitride substrate 5 , 8 . Remarkably, we find that the isospin order is only weakly perturbed, with the moiré potential catalysing the formation of topologically nontrivial gapped states whenever itinerant half- or quarter-metal states occur at half- or quarter-superlattice band filling. Our results show that rhombohedral graphene is an ideal platform for well-controlled tests of many-body theory, and reveal magnetism in moiré materials 4 , 5 , 9 , 10 to be fundamentally itinerant in nature. A study shows that rhombohedral graphene is an ideal platform for well-controlled tests of many-body theory and reveals that magnetism in moiré materials is fundamentally itinerant in nature.

Quantum wells based on mercury telluride are an experimental realization of a two-dimensional topological insulator. By using a scanning superconducting quantum interference device (SQUID) technique, the magnetic fields flowing through HgTe/CdTe heterostructures are imaged both in the quantum spin Hall and the trivial regimes, revealing the edge states associated with the quantum spin Hall state. The quantum spin Hall (QSH) state is a state of matter characterized by a non-trivial topology of its band structure, and associated conducting edge channels 1 , 2 , 3 , 4 , 5 . The QSH state was predicted 6 and experimentally demonstrated 7 to be realized in HgTe quantum wells. The existence of the edge channels has been inferred from local and non-local transport measurements in sufficiently small devices 7 , 8 , 9 . Here we directly confirm the existence of the edge channels by imaging the magnetic fields produced by current flowing in large Hall bars made from HgTe quantum wells. These images distinguish between current that passes through each edge and the bulk. On tuning the bulk conductivity by gating or raising the temperature, we observe a regime in which the edge channels clearly coexist with the conducting bulk, providing input to the question of how ballistic transport may be limited in the edge channels. Our results represent a versatile method for characterization of new QSH materials systems 10 , 11 , 12 , 13 .

The emergence of conductivity at the {001} interface of LaAlO 3 and SrTiO 3 is one of the more celebrated examples of interface engineering. Using a microscopy approach based on a sensitive magnetometry probe, it is now shown that narrow paths of enhanced conductivity occur along the crystallographic axes of the oxide structures. The ability to control materials properties through interface engineering is demonstrated by the appearance of conductivity at the interface of certain insulators, most famously the {001} interface of the band insulators LaAlO 3 and TiO 2 -terminated SrTiO 3 (STO; refs  1 , 2 ). Transport and other measurements in this system show a plethora of diverse physical phenomena 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 . To better understand the interface conductivity, we used scanning superconducting quantum interference device microscopy to image the magnetic field locally generated by current in an interface. At low temperature, we found that the current flowed in conductive narrow paths oriented along the crystallographic axes, embedded in a less conductive background. The configuration of these paths changed on thermal cycling above the STO cubic-to-tetragonal structural transition temperature, implying that the local conductivity is strongly modified by the STO tetragonal domain structure. The interplay between substrate domains and the interface provides an additional mechanism for understanding and controlling the behaviour of heterostructures.

Ferromagnetism is most common in transition metal compounds where electrons occupy highly localized d orbitals. However, ferromagnetic order may also arise in low-density two-dimensional electron systems.sup.1-5. Here we show that gate-tuned van Hove singularities in rhombohedral trilayer graphene.sup.6 drive spontaneous ferromagnetic polarization of the electron system into one or more spin and valley flavours. Using capacitance and transport measurements, we observe a cascade of transitions tuned to the density and electronic displacement field between phases in which quantum oscillations have fourfold, twofold or onefold degeneracy, associated with a spin- and valley-degenerate normal metal, spin-polarized 'half-metal', and spin- and valley-polarized 'quarter-metal', respectively. For electron doping, the salient features of the data are well captured by a phenomenological Stoner model.sup.7 that includes valley-anisotropic interactions. For hole filling, we observe a richer phase diagram featuring a delicate interplay of broken symmetries and transitions in the Fermi surface topology. Finally, we introduce a moiré superlattice using a rotationally aligned hexagonal boron nitride substrate.sup.5,8. Remarkably, we find that the isospin order is only weakly perturbed, with the moiré potential catalysing the formation of topologically nontrivial gapped states whenever itinerant half- or quarter-metal states occur at half- or quarter-superlattice band filling. Our results show that rhombohedral graphene is an ideal platform for well-controlled tests of many-body theory, and reveal magnetism in moiré materials.sup.4,5,9,10 to be fundamentally itinerant in nature.

Ferromagnetism is most common in transition metal compounds where electrons occupy highly localized d orbitals. However, ferromagnetic order may also arise in low-density two-dimensional electron systems.sup.1-5. Here we show that gate-tuned van Hove singularities in rhombohedral trilayer graphene.sup.6 drive spontaneous ferromagnetic polarization of the electron system into one or more spin and valley flavours. Using capacitance and transport measurements, we observe a cascade of transitions tuned to the density and electronic displacement field between phases in which quantum oscillations have fourfold, twofold or onefold degeneracy, associated with a spin- and valley-degenerate normal metal, spin-polarized 'half-metal', and spin- and valley-polarized 'quarter-metal', respectively. For electron doping, the salient features of the data are well captured by a phenomenological Stoner model.sup.7 that includes valley-anisotropic interactions. For hole filling, we observe a richer phase diagram featuring a delicate interplay of broken symmetries and transitions in the Fermi surface topology. Finally, we introduce a moiré superlattice using a rotationally aligned hexagonal boron nitride substrate.sup.5,8. Remarkably, we find that the isospin order is only weakly perturbed, with the moiré potential catalysing the formation of topologically nontrivial gapped states whenever itinerant half- or quarter-metal states occur at half- or quarter-superlattice band filling. Our results show that rhombohedral graphene is an ideal platform for well-controlled tests of many-body theory, and reveal magnetism in moiré materials.sup.4,5,9,10 to be fundamentally itinerant in nature.

Ferromagnetism is most common in transition metal compounds where electrons occupy highly localized d orbitals. However, ferromagnetic order may also arise in low-density two-dimensional electron systems.sup.1-5. Here we show that gate-tuned van Hove singularities in rhombohedral trilayer graphene.sup.6 drive spontaneous ferromagnetic polarization of the electron system into one or more spin and valley flavours. Using capacitance and transport measurements, we observe a cascade of transitions tuned to the density and electronic displacement field between phases in which quantum oscillations have fourfold, twofold or onefold degeneracy, associated with a spin- and valley-degenerate normal metal, spin-polarized 'half-metal', and spin- and valley-polarized 'quarter-metal', respectively. For electron doping, the salient features of the data are well captured by a phenomenological Stoner model.sup.7 that includes valley-anisotropic interactions. For hole filling, we observe a richer phase diagram featuring a delicate interplay of broken symmetries and transitions in the Fermi surface topology. Finally, we introduce a moiré superlattice using a rotationally aligned hexagonal boron nitride substrate.sup.5,8. Remarkably, we find that the isospin order is only weakly perturbed, with the moiré potential catalysing the formation of topologically nontrivial gapped states whenever itinerant half- or quarter-metal states occur at half- or quarter-superlattice band filling. Our results show that rhombohedral graphene is an ideal platform for well-controlled tests of many-body theory, and reveal magnetism in moiré materials.sup.4,5,9,10 to be fundamentally itinerant in nature.