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Lattice geometry, topological electron behaviour and the competition between different possible ground states all play a role in determining the properties of materials with a kagome lattice structure. In particular, the compounds KV 3 Sb 5 , CsV 3 Sb 5 and RbV 3 Sb 5 all feature a kagome net of vanadium atoms. These materials have recently been shown to exhibit superconductivity at low temperature and an unusual charge order at high temperature, revealing a connection to the underlying topological nature of the band structure. We highlight these discoveries, place them in the context of wider research efforts in topological physics and superconductivity, and discuss the open problems for this field. Superconductivity and ordered states formed by interactions—both of which could be unconventional—have recently been observed in a family of kagome materials.

The intricate interplay between non-trivial topology and magnetism in two-dimensional materials can lead to the emergence of interesting phenomena such as the quantum anomalous Hall effect. Here we investigate the quantum transport of both bulk crystal and exfoliated MnBi 2 Te 4 flakes in a field-effect transistor geometry. For the six septuple-layer device tuned into the insulating regime, we observe a large longitudinal resistance and zero Hall plateau, which are characteristics of an axion insulator state. The robust axion insulator state occurs in zero magnetic field, over a wide magnetic-field range and at relatively high temperatures. Moreover, a moderate magnetic field drives a quantum phase transition from the axion insulator phase to a Chern insulator phase with zero longitudinal resistance and quantized Hall resistance h / e 2 , where h is Planck’s constant and e is electron charge. Our results pave the way for using even-number septuple-layer MnBi 2 Te 4 to realize the quantized topological magnetoelectric effect and axion electrodynamics in condensed matter systems. A large longitudinal resistance and zero Hall plateau—hallmarks of an axion insulator—are found in MnBi 2 Te 4 . Moreover, a moderate magnetic field drives a quantum phase transition to a Chern insulator phase with zero longitudinal resistance and quantized Hall resistance h / e 2 .

Intertwining quantum order and non-trivial topology is at the frontier of condensed matter physics 1 – 4 . A charge-density-wave-like order with orbital currents has been proposed for achieving the quantum anomalous Hall effect 5 , 6 in topological materials and for the hidden phase in cuprate high-temperature superconductors 7 , 8 . However, the experimental realization of such an order is challenging. Here we use high-resolution scanning tunnelling microscopy to discover an unconventional chiral charge order in a kagome material, KV 3 Sb 5 , with both a topological band structure and a superconducting ground state. Through both topography and spectroscopic imaging, we observe a robust 2 × 2 superlattice. Spectroscopically, an energy gap opens at the Fermi level, across which the 2 × 2 charge modulation exhibits an intensity reversal in real space, signalling charge ordering. At the impurity-pinning-free region, the strength of intrinsic charge modulations further exhibits chiral anisotropy with unusual magnetic field response. Theoretical analysis of our experiments suggests a tantalizing unconventional chiral charge density wave in the frustrated kagome lattice, which can not only lead to a large anomalous Hall effect with orbital magnetism, but also be a precursor of unconventional superconductivity. An unconventional chiral charge order is observed in a kagome superconductor by scanning tunnelling microscopy. This charge order has unusual magnetic tunability and intertwines with electronic topology.

A three-dimensional (3D) topological insulator features a 2D surface state consisting of a single linearly dispersive Dirac cone 1 – 3 . Under broken time-reversal symmetry, the single Dirac cone is predicted to cause half-integer quantization of Hall conductance, which is a manifestation of the parity anomaly in quantum field theory 1 – 9 . However, despite various observations of quantization phenomena 10 – 15 , the half-integer quantization has not been observed because most experiments simultaneously measure a pair of equivalent Dirac cones 16 on two opposing surfaces. Here we demonstrate the half-integer quantization of Hall conductance in a synthetic heterostructure termed a semi-magnetic topological insulator, where only one surface state is gapped by magnetic doping and the opposite one is non-magnetic and gapless. We observe half-quantized Faraday and Kerr rotations with terahertz magneto-optical spectroscopy and half-quantized Hall conductance in transport at zero magnetic field. Our results suggest a condensed-matter realization of the parity anomaly 4 – 9 and open a way for studying the physics enabled by a single Dirac fermion. An electron with a linear dispersion relation should contribute half of a quantum of Hall conductance and thereby manifest the parity anomaly. This is demonstrated in a heterostructure of topological insulator materials.

Topological aspects of the electron wave function—including the Berry curvature and Chern number—play a crucial role in determining the physical properties of materials. Although the Berry curvature and its effects in materials have been studied 1 , 2 , detecting changes in the Chern number can be challenging, particularly changes in the valley Chern type. In this regard, twisted double bilayer graphene 3 – 7 has emerged as a promising platform to gain electrical control over the Berry curvature hotspots 8 and the valley Chern numbers of topological flat bands 9 , 10 . In addition, strain-induced breaking of the threefold rotation symmetry leads to a non-zero first moment of Berry curvature (called the Berry curvature dipole) 11 . Here we show that a sign change in the Berry curvature dipole detects topological transitions in the bands. In twisted double bilayer graphene, the perpendicular electric field simultaneously tunes the valley Chern number and Berry curvature dipole, providing a tunable system to probe the topological transitions. Furthermore, we find hysteresis in the transport response that is caused by switching of the electric polarization. This holds promise for next-generation Berry-curvature-based memory devices. Our technique can be emulated in three-dimensional topological systems to probe topological transitions governed by parameters such as pressure or anisotropic strain. Transport experiments highlight a technique to detect transitions in the topological state of two-dimensional materials, with possible applications in memory devices.

Magnetism typically arises from the joint effect of Fermi statistics and repulsive Coulomb interactions, which favours ground states with non-zero electron spin. As a result, controlling spin magnetism with electric fields—a longstanding technological goal in spintronics and multiferroics 1 , 2 —can be achieved only indirectly. Here we experimentally demonstrate direct electric-field control of magnetic states in an orbital Chern insulator 3 – 6 , a magnetic system in which non-trivial band topology favours long-range order of orbital angular momentum but the spins are thought to remain disordered 7 – 14 . We use van der Waals heterostructures consisting of a graphene monolayer rotationally faulted with respect to a Bernal-stacked bilayer to realize narrow and topologically non-trivial valley-projected moiré minibands 15 – 17 . At fillings of one and three electrons per moiré unit cell within these bands, we observe quantized anomalous Hall effects 18 with transverse resistance approximately equal to h /2 e 2 (where h is Planck’s constant and e is the charge on the electron), which is indicative of spontaneous polarization of the system into a single-valley-projected band with a Chern number equal to two. At a filling of three electrons per moiré unit cell, we find that the sign of the quantum anomalous Hall effect can be reversed via field-effect control of the chemical potential; moreover, this transition is hysteretic, which we use to demonstrate non-volatile electric-field-induced reversal of the magnetic state. A theoretical analysis 19 indicates that the effect arises from the topological edge states, which drive a change in sign of the magnetization and thus a reversal in the favoured magnetic state. Voltage control of magnetic states can be used to electrically pattern non-volatile magnetic-domain structures hosting chiral edge states, with applications ranging from reconfigurable microwave circuit elements to ultralow-power magnetic memories. Non-volatile electrical switching of magnetic order in an orbital Chern insulator is experimentally demonstrated using a moiré heterostructure and analysis shows that the effect is driven by topological edge states.

The recently discovered flat electronic bands and strongly correlated and superconducting phases in magic-angle twisted bilayer graphene (MATBG) 1 , 2 crucially depend on the interlayer twist angle, θ . Although control of the global θ with a precision of about 0.1 degrees has been demonstrated 1 – 7 , little information is available on the distribution of the local twist angles. Here we use a nanoscale on-tip scanning superconducting quantum interference device (SQUID-on-tip) 8 to obtain tomographic images of the Landau levels in the quantum Hall state 9 and to map the local θ variations in hexagonal boron nitride (hBN)-encapsulated MATBG devices with relative precision better than 0.002 degrees and a spatial resolution of a few moiré periods. We find a correlation between the degree of θ disorder and the quality of the MATBG transport characteristics and show that even state-of-the-art devices—which exhibit correlated states, Landau fans and superconductivity—display considerable local variation in θ of up to 0.1 degrees, exhibiting substantial gradients and networks of jumps, and may contain areas with no local MATBG behaviour. We observe that the correlated states in MATBG are particularly fragile with respect to the twist-angle disorder. We also show that the gradients of θ generate large gate-tunable in-plane electric fields, unscreened even in the metallic regions, which profoundly alter the quantum Hall state by forming edge channels in the bulk of the sample and may affect the phase diagram of the correlated and superconducting states. We thus establish the importance of θ disorder as an unconventional type of disorder enabling the use of twist-angle gradients for bandstructure engineering, for realization of correlated phenomena and for gate-tunable built-in planar electric fields for device applications. SQUID-on-tip tomographic imaging of Landau levels in magic-angle graphene provides nanoscale maps of local twist-angle disorder and shows that its properties are fundamentally different from common types of disorder.

Room-temperature realization of macroscopic quantum phases is one of the major pursuits in fundamental physics 1 , 2 . The quantum spin Hall phase 3 – 6 is a topological quantum phase that features a two-dimensional insulating bulk and a helical edge state. Here we use vector magnetic field and variable temperature based scanning tunnelling microscopy to provide micro-spectroscopic evidence for a room-temperature quantum spin Hall edge state on the surface of the higher-order topological insulator Bi 4 Br 4 . We find that the atomically resolved lattice exhibits a large insulating gap of over 200 meV, and an atomically sharp monolayer step edge hosts an in-gap gapless state, suggesting topological bulk–boundary correspondence. An external magnetic field can gap the edge state, consistent with the time-reversal symmetry protection inherent in the underlying band topology. We further identify the geometrical hybridization of such edge states, which not only supports the Z 2 topology of the quantum spin Hall state but also visualizes the building blocks of the higher-order topological insulator phase. Our results further encourage the exploration of high-temperature transport quantization of the putative topological phase reported here. We show that an insulating bulk state and helical edge state coexist in Bi 4 Br 4 and that this coexistence persists up to room temperature.

Two-dimensional electrons confined to GaAs quantum wells are hallmark platforms for probing electron–electron interactions. Many key observations have been made in these systems as sample quality has improved over the years. Here, we present a breakthrough in sample quality via source-material purification and innovation in GaAs molecular beam epitaxy vacuum chamber design. Our samples display an ultra-high mobility of 44 × 10 6  cm 2  V –1  s –1 at an electron density of 2.0 × 10 11  cm –2 . These results imply only 1 residual impurity for every 10 10 Ga/As atoms. The impact of such low impurity concentration is manifold. Robust stripe and bubble phases are observed, and several new fractional quantum Hall states emerge. Furthermore, the activation gap ( Δ ) of the fractional quantum Hall state at the Landau-level filling ( ν ) = 5/2, which is widely believed to be non-Abelian and of potential use for topological quantum computing, reaches Δ ≈ 820 mK. We expect that our results will stimulate further research on interaction-driven physics in a two-dimensional setting and substantially advance the field. Source-material purification and optimized vacuum chamber design lead to a breakthrough in GaAs sample quality.

Realizing non-trivial topological effects is challenging in acoustic systems. It is now shown that inversion symmetry breaking can be used to create acoustic analogues of the topological Haldane model. Following the discovery of the quantum Hall effect 1 , 2 and topological insulators 3 , 4 , the topological properties of classical waves began to draw attention 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 . Topologically non-trivial bands characterized by non-zero Chern numbers are realized through either the breaking of time-reversal symmetry using an external magnetic field 5 , 6 , 7 , 15 , 16 or dynamic modulation 8 , 17 . Owing to the absence of a Faraday-like effect, the breaking of time-reversal symmetry in an acoustic system is commonly realized with moving background fluids 20 , 22 , which drastically increases the engineering complexity. Here we show that we can realize effective inversion symmetry breaking and create an effective gauge flux in a reduced two-dimensional system by engineering interlayer couplings, achieving an acoustic analogue of the topological Haldane model 2 , 23 . We show that the synthetic gauge flux is closely related to Weyl points 24 , 25 , 26 in the three-dimensional band structure and the system supports chiral edge states for fixed values of k z .