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Whereas ferromagnets have been known and used for millennia, antiferromagnets were only discovered in the 1930s 1 . At large scale, because of the absence of global magnetization, antiferromagnets may seem to behave like any non-magnetic material. At the microscopic level, however, the opposite alignment of spins forms a rich internal structure. In topological antiferromagnets, this internal structure leads to the possibility that the property known as the Berry phase can acquire distinct spatial textures 2 , 3 . Here we study this possibility in an antiferromagnetic axion insulator—even-layered, two-dimensional MnBi 2 Te 4 —in which spatial degrees of freedom correspond to different layers. We observe a type of Hall effect—the layer Hall effect—in which electrons from the top and bottom layers spontaneously deflect in opposite directions. Specifically, under zero electric field, even-layered MnBi 2 Te 4 shows no anomalous Hall effect. However, applying an electric field leads to the emergence of a large, layer-polarized anomalous Hall effect of about 0.5 e 2 / h (where e is the electron charge and h is Planck’s constant). This layer Hall effect uncovers an unusual layer-locked Berry curvature, which serves to characterize the axion insulator state. Moreover, we find that the layer-locked Berry curvature can be manipulated by the axion field formed from the dot product of the electric and magnetic field vectors. Our results offer new pathways to detect and manipulate the internal spatial structure of fully compensated topological antiferromagnets 4 – 9 . The layer-locked Berry curvature represents a first step towards spatial engineering of the Berry phase through effects such as layer-specific moiré potential. A new type of Hall effect—the layer Hall effect—is produced in a 2D antiferromagnet that does not exhibit any net magnetization.

Topological phononic crystals, alike their electronic counterparts, are characterized by a bulk–edge correspondence where the interior of a material dictates the existence of stable surface or boundary modes. In the mechanical setup, such surface modes can be used for various applications such as wave guiding, vibration isolation, or the design of static properties such as stable floppy modes where parts of a system move freely. Here, we provide a classification scheme of topological phonons based on local symmetries. We import and adapt the classification of noninteracting electron systems and embed it into the mechanical setup. Moreover, we provide an extensive set of examples that illustrate our scheme and can be used to generate models in unexplored symmetry classes. Our work unifies the vast recent literature on topological phonons and paves the way to future applications of topological surface modes in mechanical metamaterials.

Applied magnetic fields underlie exotic quantum states, such as the fractional quantum Hall effect1 and Bose–Einstein condensation of spin excitations2. Superconductivity, however, is inherently antagonistic towards magnetic fields. Only in rare cases3–5 can these effects be mitigated over limited fields, leading to re-entrant superconductivity. Here, we report the coexistence of multiple high-field re-entrant superconducting phases in the spin-triplet superconductor UTe2 (ref. 6). We observe superconductivity in the highest magnetic field range identified for any re-entrant superconductor, beyond 65 T. Although the stability of superconductivity in these high magnetic fields challenges current theoretical models, these extreme properties seem to reflect a new kind of exotic superconductivity rooted in magnetic fluctuations7 and boosted by a quantum dimensional crossover8.

Berry phases and the related concept of Berry curvature can give rise to many unconventional phenomena in solids. Here, we discover a colossal orbital Zeeman effect of topological origin in a bilayer kagome metal, TbV6Sn6. Using spectroscopic imaging scanning tunnelling microscopy, we reveal that the magnetic field leads to a splitting of the gapped Dirac dispersion into two branches with enhanced momentum-dependent g factors, resulting in a substantial renormalization of the Dirac band. These measurements provide a direct observation of a magnetic field-controlled orbital Zeeman coupling to the orbital magnetic moments of up to 200 Bohr magnetons near the gapped Dirac points. Our work provides direct insight into the momentum-dependent nature of topological orbital moments and their tunability via the magnetic field, concomitant with the evolution of the spin Berry curvature. Furthermore, these results can be extended to explore large orbital magnetic moments driven by the Berry curvature governed by other quantum numbers beyond spin, such as the valley in certain graphene-based structures.

Topological phase transitions in condensed matter systems give rise to exotic states of matter such as topological insulators, superconductors, and superfluids. Photonic topological systems open a whole new realm of research and technological opportunities, exhibiting a number of important distinctions from their condensed matter counterparts. Photonic modes can leak into free space, which makes it possible to probe topological photonic phases by spectroscopic means via Fano resonances. Based on this idea, we develop a technique to retrieve the topological properties of all-dielectric metasurfaces from the measured far-field scattering characteristics. Collected angle-resolved spectra provide the momentum-dependent frequencies and lifetimes of the photonic modes that enable the retrieval of the effective Hamiltonian and extraction of the topological invariant. Our results demonstrate how the topological states of open non-Hermitian systems can be explored via far-field measurements, thus paving a way to the design of metasurfaces with unique scattering characteristics controlled via topological effects. Topological modes in photonics systems are not completely confined to the structure but can leak into free space. Here, Gorlach et al. exploit these leaky modes to probe the topological properties of a dielectric metasurface from far-field scattering measurements.

Fractional Chern insulators (FCIs) are lattice analogues of fractional quantum Hall states that may provide a new avenue towards manipulating non-Abelian excitations. Early theoretical studies 1 – 7 have predicted their existence in systems with flat Chern bands and highlighted the critical role of a particular quantum geometry. However, FCI states have been observed only in Bernal-stacked bilayer graphene (BLG) aligned with hexagonal boron nitride (hBN) 8 , in which a very large magnetic field is responsible for the existence of the Chern bands, precluding the realization of FCIs at zero field. By contrast, magic-angle twisted BLG 9 – 12 supports flat Chern bands at zero magnetic field 13 – 17 , and therefore offers a promising route towards stabilizing zero-field FCIs. Here we report the observation of eight FCI states at low magnetic field in magic-angle twisted BLG enabled by high-resolution local compressibility measurements. The first of these states emerge at 5 T, and their appearance is accompanied by the simultaneous disappearance of nearby topologically trivial charge density wave states. We demonstrate that, unlike the case of the BLG/hBN platform, the principal role of the weak magnetic field is merely to redistribute the Berry curvature of the native Chern bands and thereby realize a quantum geometry favourable for the emergence of FCIs. Our findings strongly suggest that FCIs may be realized at zero magnetic field and pave the way for the exploration and manipulation of anyonic excitations in flat moiré Chern bands. A study using local compressibility measurements reports fractional Chern insulator states at low magnetic field in magic-angle twisted bilayer graphene, and establishes the applied magnetic field as a means to tune the Berry curvature distribution.

Electron correlations often lead to emergent orders in quantum materials, and one example is the kagome lattice materials where topological states exist in the presence of strong correlations between electrons. This arises from the features of the electronic band structure that are associated with the kagome lattice geometry: flat bands induced by destructive interference of the electronic wavefunctions, topological Dirac crossings and a pair of van Hove singularities. Various correlated electronic phases have been discovered in kagome lattice materials, including magnetism, charge density waves, nematicity and superconductivity. Recently, a charge density wave was discovered in the magnetic kagome FeGe, providing a platform for understanding the interplay between charge order and magnetism in kagome materials. Here we observe all three electronic signatures of the kagome lattice in FeGe using angle-resolved photoemission spectroscopy. The presence of van Hove singularities near the Fermi level is driven by the underlying magnetic exchange splitting. Furthermore, we show spectral evidence for the charge density wave as gaps near the Fermi level. Our observations point to the magnetic interaction-driven band modification resulting in the formation of the charge density wave and indicate an intertwined connection between the emergent magnetism and charge order in this moderately correlated kagome metal.The observation of band structure features typical of the kagome lattice in FeGe suggests that an interplay of magnetism and electronic correlations determines the physics of this material.

Broadband, efficient and fast conversion of light to electricity is crucial for sensing and clean energy. The bulk photovoltaic effect (BPVE) is a second-order nonlinear optical effect that intrinsically converts light into electrical current. Here, we demonstrate a large mid-infrared BPVE in microscopic devices of the Weyl semimetal TaAs. This discovery results from combining recent developments in Weyl semimetals, focused-ion beam fabrication and theoretical works suggesting a connection between BPVE and topology. We also present a detailed symmetry analysis that allows us to separate the shift current response from photothermal effects. The magnitude and wavelength range of the assigned shift current may impact optical detectors, clean energy and topology, and demonstrate the utility of Weyl semimetals for practical applications.A large bulk photovoltaic effect is observed in the type-I Weyl semimetal TaAs, and attributed to the diverging Berry curvature of the Weyl nodes.

A kagome lattice of 3 d transition metal ions is a versatile platform for correlated topological phases hosting symmetry-protected electronic excitations and magnetic ground states. However, the paradigmatic states of the idealized two-dimensional kagome lattice—Dirac fermions and flat bands—have not been simultaneously observed. Here, we use angle-resolved photoemission spectroscopy and de Haas–van Alphen quantum oscillations to reveal coexisting surface and bulk Dirac fermions as well as flat bands in the antiferromagnetic kagome metal FeSn, which has spatially decoupled kagome planes. Our band structure calculations and matrix element simulations demonstrate that the bulk Dirac bands arise from in-plane localized Fe-3 d orbitals, and evidence that the coexisting Dirac surface state realizes a rare example of fully spin-polarized two-dimensional Dirac fermions due to spin-layer locking in FeSn. The prospect to harness these prototypical excitations in a kagome lattice is a frontier of great promise at the confluence of topology, magnetism and strongly correlated physics. A prototypical kagome metal with magnetic and topological properties is identified.

The recent discovery of novel charge density wave (CDW) and pair density wave (PDW) in kagomé lattice superconductors A V 3 Sb 5 ( A  = K, Rb, Cs) hints at unexpected time-reversal symmetry breaking correlated and topological states whose physical origin and broader implications are not understood. Here, we make conceptual advances toward a mechanism behind the striking observations and new predictions for novel macroscopic phase coherent quantum states. We show that the metallic CDW state with circulating loop currents is a doped orbital Chern insulator near van Hove filling. The emergent Chern Fermi pockets (CFPs) carry concentrated Berry curvature and orbital magnetic moment. We find that the pairing of electrons on the CFPs leads to a superconducting state with an emergent vortex-antivortex lattice and the formation of a complex triple- Q PDW. A plethora of correlated and topological states emerge, including a never-before-encountered chiral topological PDW superconductor, a loop-current pseudogap phase, and vestigial charge-4 e and charge-6 e superconductivity in staged melting of the vortex-antivortex lattice and hexatic liquid crystal. Our findings reveal previously unknown nature of the superconducting state of a current-carrying Chern metal, with broad implications for correlated and topological materials. The recent discovery of novel charge density wave (CDW) and pair density wave (PDW) in kagome superconductors (SC) A V 3 Sb 5 motivates theoretical study of these phenomena. Here, the authors propose that the CDW state is an orbital Chern metal, leading to a SC state with a chiral PDW, the melting of which leads to vestigial electronic orders including charge-4e and 6e SC.