Explore the vast range of titles available.

Recent experiments revealed that Mn3Sn and Mn3Ge exhibit a strong anomalous Hall effect at room temperature, provoking us to explore their electronic structures for topological properties. By ab initio band structure calculations, we have observed the existence of multiple Weyl points in the bulk and corresponding Fermi arcs on the surface, predicting antiferromagnetic Weyl semimetals in Mn3Ge and Mn3Sn. Here the chiral antiferromagnetism in the Kagome-type lattice structure is essential to determine the positions and numbers of Weyl points. Our work further reveals a new guiding principle to search for magnetic Weyl semimetals among materials that exhibit a strong anomalous Hall effect.

Of the two stable forms of graphite, hexagonal and rhombohedral, the former is more common and has been studied extensively. The latter is less stable, which has so far precluded its detailed investigation, despite many theoretical predictions about the abundance of exotic interaction-induced physics 1 – 6 . Advances in van der Waals heterostructure technology 7 have now allowed us to make high-quality rhombohedral graphite films up to 50 graphene layers thick and study their transport properties. Here we show that the bulk electronic states in such rhombohedral graphite are gapped 8 and, at low temperatures, electron transport is dominated by surface states. Because of their proposed topological nature, the surface states are of sufficiently high quality to observe the quantum Hall effect, whereby rhombohedral graphite exhibits phase transitions between a gapless semimetallic phase and a gapped quantum spin Hall phase with giant Berry curvature. We find that an energy gap can also be opened in the surface states by breaking their inversion symmetry by applying a perpendicular electric field. Moreover, in rhombohedral graphite thinner than four nanometres, a gap is present even without an external electric field. This spontaneous gap opening shows pronounced hysteresis and other signatures characteristic of electronic phase separation, which we attribute to emergence of strongly correlated electronic surface states. High-quality rhombohedral graphite films are found to offer an alternative to twisted bilayer graphene as a platform for studying correlated physics in carbon materials.

The fractional quantum anomalous Hall effect (FQAHE), the analogue of the fractional quantum Hall effect 1 at zero magnetic field, is predicted to exist in topological flat bands under spontaneous time-reversal-symmetry breaking 2 – 6 . The demonstration of FQAHE could lead to non-Abelian anyons that form the basis of topological quantum computation 7 – 9 . So far, FQAHE has been observed only in twisted MoTe 2 at a moiré filling factor v  > 1/2 (refs. 10 – 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 R x y = h v e 2 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 R xx . R xy equals 2 h e 2 at v  = 1/2 and varies linearly with v , similar to the composite Fermi liquid in the half-filled lowest Landau level at high magnetic fields 14 – 16 . By tuning the gate-displacement field D and v , we observed phase transitions from composite Fermi liquid 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 field 7 – 9 , 17 – 19 , especially considering a lateral junction between FQAHE and superconducting regions in the same device 20 – 22 . Integer and fractional quantum anomalous Hall effects in a rhombohedral pentalayer graphene–hBN moiré superlattice are observed, providing an ideal platform for exploring charge fractionalization and (non-Abelian) anyonic braiding at zero magnetic field.

The integer quantum anomalous Hall (QAH) effect is a lattice analogue of the quantum Hall effect at zero magnetic field 1 – 3 . This phenomenon occurs in systems with topologically non-trivial bands and spontaneous time-reversal symmetry breaking. Discovery of its fractional counterpart in the presence of strong electron correlations, that is, the fractional QAH effect 4 – 7 , would open a new chapter in condensed matter physics. Here we report the direct observation of both integer and fractional QAH effects in electrical measurements on twisted bilayer MoTe 2 . At zero magnetic field, near filling factor ν  = −1 (one hole per moiré unit cell), we see an integer QAH plateau in the Hall resistance R xy quantized to h / e 2  ± 0.1%, whereas the longitudinal resistance R xx vanishes. Remarkably, at ν   =  −2/3 and −3/5, we see plateau features in R xy at 3 2 h / e 2 ± 1 % and 5 3 h / e 2 ± 3 % , respectively, whereas R xx remains small. All features shift linearly versus applied magnetic field with slopes matching the corresponding Chern numbers −1, −2/3 and −3/5, precisely as expected for integer and fractional QAH states. Additionally, at zero magnetic field, R xy is approximately 2 h / e 2 near half-filling ( ν   = −1/2) and varies linearly as ν   is tuned. This behaviour resembles that of the composite Fermi liquid in the half-filled lowest Landau level of a two-dimensional electron gas at high magnetic field 8 – 14 . Direct observation of the fractional QAH and associated effects enables research in charge fractionalization and anyonic statistics at zero magnetic field. Transport measurements in twisted bilayer MoTe 2 reveal quantized Hall resistance plateaus and composite Fermi liquid-like behaviour under zero magnetic field, constituting a direct observation of integer and fractional quantum anomalous Hall effects.

Magnetic topological materials represent a class of compounds with properties that are strongly influenced by the topology of their electronic wavefunctions coupled with the magnetic spin configuration. Such materials can support chiral electronic channels of perfect conduction, and can be used for an array of applications, from information storage and control to dissipationless spin and charge transport. Here we review the theoretical and experimental progress achieved in the field of magnetic topological materials, beginning with the theoretical prediction of the quantum anomalous Hall effect without Landau levels, and leading to the recent discoveries of magnetic Weyl semimetals and antiferromagnetic topological insulators. We outline recent theoretical progress that has resulted in the tabulation of, for the first time, all magnetic symmetry group representations and topology. We describe several experiments realizing Chern insulators, Weyl and Dirac magnetic semimetals, and an array of axionic and higher-order topological phases of matter, and we survey future perspectives. Recent theoretical and experimental progress in identifying and understanding magnetic topological materials is reviewed, highlighting the antiferromagnetic topological insulator MnBi 2 Te 4 and the ferromagnetic Weyl semimetal Co 3 Sn 2 S 2 , and future research directions are discussed.

The interplay between spontaneous symmetry breaking and topology can result in exotic quantum states of matter. A celebrated example is the quantum anomalous Hall (QAH) state, which exhibits an integer quantum Hall effect at zero magnetic field owing to intrinsic ferromagnetism 1 – 3 . In the presence of strong electron–electron interactions, fractional QAH (FQAH) states at zero magnetic field can emerge 4 – 8 . These states could host fractional excitations, including non-Abelian anyons—crucial building blocks for topological quantum computation 9 . Here we report experimental signatures of FQAH states in a twisted molybdenum ditelluride (MoTe 2 ) bilayer. Magnetic circular dichroism measurements reveal robust ferromagnetic states at fractionally hole-filled moiré minibands. Using trion photoluminescence as a sensor 10 , we obtain a Landau fan diagram showing linear shifts in carrier densities corresponding to filling factor v  = −2/3 and v  = −3/5 ferromagnetic states with applied magnetic field. These shifts match the Streda formula dispersion of FQAH states with fractionally quantized Hall conductance of σ x y = − 2 3 e 2 h and σ x y = − 3 5 e 2 h , respectively. Moreover, the v  = −1 state exhibits a dispersion corresponding to Chern number −1, consistent with the predicted QAH state 11 – 14 . In comparison, several non-ferromagnetic states on the electron-doping side do not disperse, that is, they are trivial correlated insulators. The observed topological states can be electrically driven into topologically trivial states. Our findings provide evidence of the long-sought FQAH states, demonstrating MoTe 2 moiré superlattices as a platform for exploring fractional excitations. Signatures of fractional quantum anomalous Hall states at zero magnetic field are observed in a fractionally filled moiré superlattice in a molybdenum ditelluride twisted bilayer.

The Berry curvature and quantum metric are the imaginary part and real part, respectively, of the quantum geometric tensor, which characterizes the topology of quantum states 1 . The Berry curvature is known to generate a number of important transport phenomena, such as the quantum Hall effect and the anomalous Hall effect 2 , 3 ; however, the consequences of the quantum metric have rarely been probed by transport measurements. Here we report the observation of quantum-metric-induced nonlinear transport, including both a nonlinear anomalous Hall effect and a diode-like non-reciprocal longitudinal response, in thin films of a topological antiferromagnet, MnBi 2 Te 4 . Our observations reveal that the transverse and longitudinal nonlinear conductivities reverse signs when reversing the antiferromagnetic order, diminish above the Néel temperature and are insensitive to disorder scattering, thus verifying their origin in the band-structure topology. They also flip signs between electron- and hole-doped regions, in agreement with theoretical calculations. Our work provides a means to probe the quantum metric through nonlinear transport and to design magnetic nonlinear devices. Quantum-metric-induced nonlinear transport, including the nonlinear anomalous Hall effect and a diode-like response, is observed in thin films of a topological antiferromagnet, providing a means to design magnetic nonlinear devices.

Quantum spin Hall (QSH) insulators are two-dimensional electronic materials that have a bulk band gap similar to an ordinary insulator but have topologically protected pairs of edge modes of opposite chiralities 1 , 2 , 3 , 4 , 5 – 6 . So far, experimental studies have found only integer QSH insulators with counter-propagating up-spins and down-spins at each edge leading to a quantized conductance G 0  =  e 2 / h (with e and h denoting the electron charge and Planck’s constant, respectively) 7 , 8 , 9 , 10 , 11 , 12 , 13 – 14 . Here we report transport evidence of a fractional QSH insulator in 2.1° twisted bilayer MoTe 2 , which supports spin- S z conservation and flat spin-contrasting Chern bands 15 , 16 . At filling factor ν  = 3 of the moiré valence bands, each edge contributes a conductance 3 2 G 0 with zero anomalous Hall conductivity. The state is probably a time-reversal pair of the even-denominator 3/2-fractional Chern insulators. Furthermore, at ν   =  2, 4 and 6, we observe a single, double and triple QSH insulator with each edge contributing a conductance G 0 , 2 G 0 and 3 G 0 , respectively. Our results open up the possibility of realizing time-reversal symmetric non-abelian anyons and other unexpected topological phases in highly tunable moiré materials 17 , 18 – 19 . Transport evidence of a fractional quantum spin Hall insulator is reported in 2.1°-twisted bilayer MoTe 2 , which supports spin- S z conservation and flat spin-contrasting Chern bands.

Moiré superlattices 1 , 2 have recently emerged as a platform upon which correlated physics and superconductivity can be studied with unprecedented tunability 3 – 6 . Although correlated effects have been observed in several other moiré systems 7 – 17 , magic-angle twisted bilayer graphene remains the only one in which robust superconductivity has been reproducibly measured 4 – 6 . Here we realize a moiré superconductor in magic-angle twisted trilayer graphene (MATTG) 18 , which has better tunability of its electronic structure and superconducting properties than magic-angle twisted bilayer graphene. Measurements of the Hall effect and quantum oscillations as a function of density and electric field enable us to determine the tunable phase boundaries of the system in the normal metallic state. Zero-magnetic-field resistivity measurements reveal that the existence of superconductivity is intimately connected to the broken-symmetry phase that emerges from two carriers per moiré unit cell. We find that the superconducting phase is suppressed and bounded at the Van Hove singularities that partially surround the broken-symmetry phase, which is difficult to reconcile with weak-coupling Bardeen–Cooper–Schrieffer theory. Moreover, the extensive in situ tunability of our system allows us to reach the ultrastrong-coupling regime, characterized by a Ginzburg–Landau coherence length that reaches the average inter-particle distance, and very large T BKT / T F values, in excess of 0.1 (where T BKT and T F are the Berezinskii–Kosterlitz–Thouless transition and Fermi temperatures, respectively). These observations suggest that MATTG can be electrically tuned close to the crossover to a two-dimensional Bose–Einstein condensate. Our results establish a family of tunable moiré superconductors that have the potential to revolutionize our fundamental understanding of and the applications for strongly coupled superconductivity. Highly tunable moiré superconductivity is observed in magic-angle twisted trilayer graphene, and observations suggest that this superconductor can be tuned close to the crossover to a two-dimensional Bose–Einstein condensate.

Insulating states can be topologically nontrivial, a well-established notion that is exemplified by the quantum Hall effect and topological insulators. By contrast, topological metals have not been experimentally evidenced until recently. In systems with strong correlations, they have yet to be identified. Heavy-fermion semimetals are a prototype of strongly correlated systems and, given their strong spin-orbit coupling, present a natural setting to make progress. Here, we advance a Weyl–Kondo semimetal phase in a periodic Anderson model on a noncentrosymmetric lattice. The quasiparticles near the Weyl nodes develop out of the Kondo effect, as do the surface states that feature Fermi arcs. We determine the key signatures of this phase, which are realized in the heavy-fermion semimetal Ce₃Bi₄Pd₃. Our findings provide the much-needed theoretical foundation for the experimental search of topological metals with strong correlations and open up an avenue for systematic studies of such quantum phases that naturally entangle multiple degrees of freedom.