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Altermagnetism represents an emergent collinear magnetic phase with compensated order and an unconventional alternating even-parity wave spin order in the non-relativistic band structure. We investigate directly this unconventional band splitting near the Fermi energy through spin-integrated soft X-ray angular resolved photoemission spectroscopy. The experimentally obtained angle-dependent photoemission intensity, acquired from epitaxial thin films of the predicted altermagnet CrSb, demonstrates robust agreement with the corresponding band structure calculations. In particular, we observe the distinctive splitting of an electronic band on a low-symmetry path in the Brilliouin zone that connects two points featuring symmetry-induced degeneracy. The measured large magnitude of the spin splitting of approximately 0.6 eV and the position of the band just below the Fermi energy underscores the significance of altermagnets for spintronics based on robust broken time reversal symmetry responses arising from exchange energy scales, akin to ferromagnets, while remaining insensitive to external magnetic fields and possessing THz dynamics, akin to antiferromagnets. The fundamental hallmark of altermagnetism lies in the spin splitting of electronic valence bands. Here, the authors observe splitting in metallic CrSb, revealing an exceptionally large value and energetic placement just below the Fermi energy.

The crystal structure of a material creates a periodic potential that electrons move through giving rise to its electronic band structure. When two-dimensional materials are stacked, the resulting moiré pattern introduces an additional periodicity so that the twist angle between the layers becomes an extra degree of freedom for the resulting heterostructure. As this angle changes, the electronic band structure is modified leading to the possibility of flat bands with localized states and enhanced electronic correlations 1 – 6 . In transition metal dichalcogenides, flat bands have been theoretically predicted to occur for long moiré wavelengths over a range of twist angles around 0° and 60° (ref. 4 ) giving much wider versatility than magic-angle twisted bilayer graphene. Here, we show the existence of a flat band in the electronic structure of 3° and 57.5° twisted bilayer WSe 2 samples using scanning tunnelling spectroscopy. Our direct spatial mapping of wavefunctions at the flat-band energy show that the localization of the flat bands is different for 3° and 57.5°, in agreement with first-principles density functional theory calculations 4 . Using scanning tunnelling spectroscopy, the flat bands in twisted bilayer WSe 2 are shown near both 0° and 60° twist angles.

Recent series of theoretical and experimental reports have driven attention to time-reversal symmetry-breaking spintronic and spin-splitting phenomena in materials with collinear-compensated magnetic order incompatible with conventional ferromagnetism or antiferromagnetism. Here we employ an approach based on nonrelativistic spin-symmetry groups that resolves the conflicting notions of unconventional ferromagnetism or antiferromagnetism by delimiting a third basic collinear magnetic phase. We derive that all materials hosting this collinear-compensated magnetic phase are characterized by crystal-rotation symmetries connecting opposite-spin sublattices separated in the real space and opposite-spin electronic states separated in the momentum space. We describe prominent extraordinary characteristics of the phase, including the alternating spin-splitting sign and broken time-reversal symmetry in the nonrelativistic band structure, the planar or bulkd-,g-, ori-wave symmetry of the spin-dependent Fermi surfaces, spin-degenerate nodal lines and surfaces, band anisotropy of individual spin channels, and spin-split general, as well as time-reversal invariant momenta. Guided by the spin-symmetry principles, we discover in ab initio calculations outlier materials with an extraordinary nonrelativistic spin splitting, whose eV-scale and momentum dependence are determined by the crystal potential of the nonmagnetic phase. This spin-splitting mechanism is distinct from conventional relativistic spin-orbit coupling and ferromagnetic exchange, as well as from the previously considered anisotropic exchange mechanism in compensated magnets. Our results, combined with our identification of material candidates for the phase ranging from insulators and metals to a parent crystal of cuprate superconductors, underpin research of novel quantum phenomena and spintronic functionalities in high-temperature magnets with light elements, vanishing net magnetization, and strong spin coherence. In the discussion, we argue that the conflicting notions of unconventional ferromagnetism or antiferromagnetism, on the one hand, and our symmetry-based delimitation of the third phase, on the other hand, favor a distinct term referring to the phase. The alternating spin polarizations in both the real-space crystal structure and the momentum-space band structure characteristic of this unconventional magnetic phase suggest a term altermagnetism. We point out thatd-wave altermagnetism represents a realization of the long-sought-after counterpart in magnetism of the unconventionald-wave superconductivity.

Adding irregularity to a system can lead to a transition from a more orderly to a less orderly phase. Meier et al. demonstrated a counterintuitive transition in the opposite direction: Controlled fluctuations in the system's parameters caused it to become topologically nontrivial. The starting point was a one-dimensional lattice of ultracold rubidium atoms in momentum space whose band structure was topologically trivial. The researchers then introduced fluctuations in the tunneling between the lattice sites and monitored the atomic “wires” as the amplitude of the fluctuations increased. The wires first became topologically nontrivial and then went back to trivial for sufficient disorder strengths. Science , this issue p. 929 Controlled fluctuations in the tunneling between the sites of an atomic wire in momentum space cause a topological transition. Topology and disorder have a rich combined influence on quantum transport. To probe their interplay, we synthesized one-dimensional chiral symmetric wires with controllable disorder via spectroscopic Hamiltonian engineering, based on the laser-driven coupling of discrete momentum states of ultracold atoms. Measuring the bulk evolution of a topological indicator after a sudden quench, we observed the topological Anderson insulator phase, in which added disorder drives the band structure of a wire from topologically trivial to nontrivial. In addition, we observed the robustness of topologically nontrivial wires to weak disorder and measured the transition to a trivial phase in the presence of strong disorder. Atomic interactions in this quantum simulation platform may enable realizations of strongly interacting topological fluids.

Electron–electron interactions play an important role in graphene and related systems and can induce exotic quantum states, especially in a stacked bilayer with a small twist angle 1 – 7 . For bilayer graphene where the two layers are twisted by the ‘magic angle’, flat band and strong many-body effects lead to correlated insulating states and superconductivity 4 – 7 . In contrast to monolayer graphene, the band structure of untwisted bilayer graphene can be further tuned by a displacement field 8 – 10 , providing an extra degree of freedom to control the flat band that should appear when two bilayers are stacked on top of each other. Here, we report the discovery and characterization of displacement field-tunable electronic phases in twisted double bilayer graphene. We observe insulating states at a half-filled conduction band in an intermediate range of displacement fields. Furthermore, the resistance gap in the correlated insulator increases with respect to the in-plane magnetic fields and we find that the g factor, according to the spin Zeeman effect, is ~2, indicating spin polarization at half-filling. These results establish twisted double bilayer graphene as an easily tunable platform for exploring quantum many-body states. Placing two Bernal-stacked graphene bilayers on top of each other with a small twist angle gives correlated states. As the band structure can be tuned by an electric field, this platform is a more varied setting to study correlated electrons.

Solid electrolytes hold great promise for enabling the use of Li metal anodes. The main problem is that during cycling, Li can infiltrate along grain boundaries and cause short circuits, resulting in potentially catastrophic battery failure. At present, this phenomenon is not well understood. Here, through electron microscopy measurements on a representative system, Li 7 La 3 Zr 2 O 12 , we discover that Li infiltration in solid oxide electrolytes is strongly associated with local electronic band structure. About half of the Li 7 La 3 Zr 2 O 12 grain boundaries were found to have a reduced bandgap, around 1–3 eV, making them potential channels for leakage current. Instead of combining with electrons at the cathode, Li + ions are hence prematurely reduced by electrons at grain boundaries, forming local Li filaments. The eventual interconnection of these filaments results in a short circuit. Our discovery reveals that the grain-boundary electronic conductivity must be a primary concern for optimization in future solid-state battery design. Solid electrolytes are promising for enabling the use of Li metal anodes but Li infiltration along grain boundaries can lead to battery failure. Li infiltration in a model solid oxide electrolyte is now found to be strongly associated with local electronic band structure.

Black phosphorus consists of stacked layers of phosphorene, a two-dimensional semiconductor with promising device characteristics. We report the realization of a widely tunable band gap in few-layer black phosphorus doped with potassium using an in situ surface doping technique. Through band structure measurements and calculations, we demonstrate that a vertical electric field from dopants modulates the band gap, owing to the giant Stark effect, and tunes the material from a moderate-gap semiconductor to a band-inverted semimetal. At the critical field of this band inversion, the material becomes a Dirac semimetal with anisotropic dispersion, linear in armchair and quadratic in zigzag directions. The tunable band structure of black phosphorus may allow great flexibility in design and optimization of electronic and optoelectronic devices.

The study of spatial symmetries was accomplished during the last century and had greatly improved our understanding of the properties of solids. Nowadays, the symmetry data of any crystal can be readily extracted from standard first-principles calculation. On the other hand, the topological data (topological invariants), the defining quantities of nontrivial topological states, are in general considerably difficult to obtain, and this difficulty has critically slowed down the search for topological materials. Here we provide explicit and exhaustive mappings from symmetry data to topological data for arbitrary gapped band structure in the presence of time-reversal symmetry and any one of the 230 space groups. The mappings are completed using the theoretical tools of layer construction and symmetry-based indicators. With these results, finding topological invariants in any given gapped band structure reduces to a simple search in the mapping tables provided. The defining quantities of topological materials—the topological invariants—are often difficult to calculate. Here, Song et al. report a simplified method to calculate both the symmetry data and the topological invariants for arbitrary gapped band structure with time-reversal symmetry.

The application of strain to semiconductors allows for controlled modification of their band structure. This principle is employed for the manufacturing of devices ranging from high-performance transistors to solid-state lasers. Traditionally, strain is typically achieved via growth on lattice-mismatched substrates. For two-dimensional (2D) semiconductors, this is not feasible as they typically do not interact epitaxially with the substrate. Here, we demonstrate controlled strain engineering of 2D semiconductors during synthesis by utilizing the thermal coefficient of expansion mismatch between the substrate and semiconductor. Using WSe 2 as a model system, we demonstrate stable built-in strains ranging from 1% tensile to 0.2% compressive on substrates with different thermal coefficient of expansion. Consequently, we observe a dramatic modulation of the band structure, manifested by a strain-driven indirect-to-direct bandgap transition and brightening of the dark exciton in bilayer and monolayer WSe 2 , respectively. The growth method developed here should enable flexibility in design of more sophisticated devices based on 2D materials. Strain engineering is an essential tool for modifying local electronic properties in silicon-based electronics. Here, Ahn et al. demonstrate control of biaxial strain in two-dimensional materials based on the growth substrate, enabling more complex low-dimensional electronics.

Emergent phenomena arising from the collective behaviour of electrons is expected when Coulomb interactions dominate over the kinetic energy, and one way to create this situation is to reduce the electronic bandwidth. Bernal-stacked bilayer graphene intrinsically supports saddle points in the band structure that are predicted to host a variety of spontaneous symmetry-broken states 1 – 7 . Here we show that bilayer graphene displays a cascade of symmetry-broken states with spontaneous spin and valley isospin ordering at zero magnetic field. We independently tune the carrier density and electric displacement field to explore the phase space of isospin order. Itinerant ferromagnetic states emerge near the conduction and valence band edges with complete spin and valley polarization. At larger hole densities, twofold degenerate quantum oscillations manifest in an additional symmetry-broken state that is enhanced by the application of an in-plane magnetic field. Both symmetry-broken states display enhanced layer polarization, suggesting a coupling to the layer degree of freedom 1 , 7 . These states occur in the absence of a moiré superlattice and are intrinsic to natural graphene bilayers. Therefore, we demonstrate that bilayer graphene represents a related but distinct approach to produce collective behaviour from flat dispersion, complementary to engineered moiré structures. The flat portions of the band structure in bilayer graphene are shown to support interaction-driven symmetry-broken states, similar to moiré heterostructures.