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The energy–momentum relationship of certain fermions resembles an hourglass, which is movable but unremovable; this robust property follows from the intertwining of spatial symmetries with the band theory of crystals, revised with mathematical connections to topology and cohomology.

Fractional Chern insulators have been observed in moiré MoTe₂ at zero magnetic field, but the expected zero longitudinal resistance has not been demonstrated. Now it is shown that improving device quality allows this effect to appear.

Solid-state materials have emerged as a platform for probing and manipulating topological phases of matter. This Review surveys topological materials discovery in nonmagnetic crystalline solids, focusing on the role of crystal symmetry and geometry in topological material predictions.

Fractional Chern insulators, and their time-reversal analogs, fractional topological insulators, are realizations of topological order in flat-band electronic systems; while the former have been realized experimentally in twisted bilayer MoTe₂, the latter have remained more elusive. Here, using exact diagonalization calculations, the authors propose routes towards engineering fractional topological insulators in twisted bilayer MoTe₂ and other moiré materials.

The authors identify a novel delocalization mechanism for topologically trivial obstructed insulators. In transitioning from two topologically trivial states, where one would expect Anderson’s localization to take place, a delocalized ‘critical metal phase’ appears.

Majorana fermions, particles which are their own antiparticles, are predicted to exist in systems combining superconductivity and topologically non-trivial band structure. Here, the authors propose means to create and manipulate such excitations in one-dimensional chains of adatoms on superconducting surfaces.

Strong electron–electron interactions in magic-angle twisted bilayer graphene can fundamentally change the topology of the system’s flat bands, producing a hierarchy of strongly correlated topological insulators in modest magnetic fields.

Many topological crystalline phases have unknown physical responses. Here, the authors systematically extend the theory of defect and flux responses to predict zero-dimensional (0D) states in topological crystalline materials, including 2D PbTe monolayers and 3D SnTe.

Chiral p-wave superconductors may form the basis for topological quantum computers however one has yet to be identified. Here, the authors propose that such a state may be created by a two-dimensional array of magnetic adatoms on the surface of a superconductor with strong spin-orbit coupling.

A proposed theoretical explanation for the electronic behaviour of moiré graphene is the coexistence of light and heavy electrons. Now local thermoelectric measurements hint that this model could be accurate.

The conducting surface states of 3D topological insulators are two-dimensional. In an analogous way, the edge states of 2D topological insulators are one-dimensional. Direct evidence of this one-dimensionality is now presented, by means of scanning tunnelling spectroscopy, for bismuth bilayers—one of the first theoretically predicted 2D topological insulators.

A new class of moiré materials based on monolayers with triangular lattices and low-energy states at the M points of the Brillouin zone is introduced, demonstrating emergent momentum-space non-symmorphic symmetries, a kagome plane-wave lattice structure, and potential quasi-one-dimensionality.

The band topology of nonmagnetic crystals can be characterized by Topological Quantum Chemistry (TQC), whereas the band topology of magnetic crystals remains unexplored. Here, the authors extend TQC to the magnetic space groups to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids.

A catalogue of the naturally occurring three-dimensional stoichiometric materials with flat bands around the Fermi level provides a powerful search engine for future theoretical and experimental studies.

Interacting electrons in Hofstadter bands can form symmetry-broken topological states. These are now revealed in magic-angle twisted bilayer graphene, and their properties are influenced by non-uniform quantum geometry.

Condensed-matter physics brings us quasiparticles that behave like massless fermions.

Recent theoretical work suggested that magic-angle graphene can simultaneously host localized and itinerant electrons. Now thermoelectric measurements provide evidence of this.

Quantum geometry and electron–phonon coupling are two fundamental concepts in condensed matter physics that govern many correlated ground states. Now a generalized theory connects these two ideas.

The study of the band structure and crystal symmetry of the semimetal bismuth indicates that this material is a higher-order topological insulator hosting robust one-dimensional metallic states on the hinges of the crystal.

Twisted double bilayer graphene is predicted to be a topological insulator under certain conditions. Simultaneous bulk and edge measurements now show metallic transport with a bulk bandgap, suggestive of this prediction.

The mechanism of the multiple-q charge density wave phase in the antiferromagnetic kagome metal FeGe is not fully understood. Here the authors reveal dimerization-driven hexagonal charge-diffuse precursor and identify the fraction of dimerized/undimerized states as the key order parameter of the phase transition.

In magic-angle twisted bilayer graphene, topological Chern bands that are driven by electron–electron interactions appear at all the integer fillings of the moiré unit cell. The Rashba-like higher-energy bands also show Landau-level crossings.

The recently discovered charge density wave in ScV₆Sn₆ kagome metal is under intense debate. By using a combination of experimental and theoretical techniques, the authors point to the role of flat phonon mode softening and momentum-dependent electron-phonon coupling in the formation of the charge density wave.

The recently-developed topological heavy fermion model explains the low energy electrons of magic-angle twisted bilayer graphene as a hybridization between states localized at AA stacking sites and itinerant topological states, denoted by f and c electrons in analogy to heavy fermion systems. Here, the authors extend this model to a nonzero magnetic field, obtaining interacting Hofstadter spectra in the flatband limit by analytic methods.

A general theoretical technique is introduced to identify materials that host flat bands. Applying topological quantum chemistry provides the generating bases for these flat bands in all space groups.

In analogy with quantum Hall systems, it may be possible to find non-abelian anyons in the higher bands of Chern insulators. Now, the phase diagram of the second moiré band of twisted MoTe₂ is explored, laying the groundwork for such investigations.

Strong electron–electron interactions create a charge-density wave that modifies the topological state of the Weyl semimetal (TaSe₄)₂I. This implies the possibility of experimentally simulating axion electrodynamics in a solid-state material.

The concept of quasi-symmetry—a perturbatively small deviation from exact symmetry—is introduced and leads to topological materials with strong resilience to perturbations.

The existence of a topological bulk-boundary correspondence for Dirac semimetals has remained an open question. Here, Wieder et al. predict one-dimensional hinge states originating from bulk three-dimensional Dirac points in solid-state Dirac semimetals, revealing condensed matter Dirac fermions to be higher-order topological.

Angle-resolved photoemission spectroscopy of superconducting magic-angle twisted bilayer graphene reveals flat-band replicas that are indicative of strong electron–phonon coupling; these replicas are absent in non-superconducting twisted bilayer graphene.

A new type of topological semimetal is described, which contains so-called type-II Weyl fermions and has very different properties to standard Weyl semimetals, owing to the existence of an open Fermi surface rather than a point-like one in the vicinity of Weyl points; WTe₂ is predicted to be one such semimetal.

High-throughput calculations are performed to predict approximately 130 magnetic topological materials, with complete electronic structure calculations and topological phase diagrams.

Topological quantum chemistry and newly developed codes are used to analyse and compute the topological properties of materials in a large crystal database and to identify new topological phases, finding that more than 27 per cent of all materials in nature are topological.

A complete electronic band theory is presented that describes the global properties of all possible band structures and materials, and can be used to predict new topological insulators and semimetals.

Moiré patterns have been experimentally observed in heterostructures comprised of topological insulator films. Here, the authors propose that topological insulator-based moiré heterostructures could be a host of isolated topologically non-trivial moiré minibands for the study of the interplay between topology and correlation.

While the classification of single-particle topological phases has been established, recent efforts have been made to extend it to interacting limit. Here the authors present a classification of interacting topological systems in 2D based on the generalization of real space invariants.

High-resolution scanning tunnelling microscopy is used to observe the quantum textures of the many-body wavefunctions of the correlated insulating, pseudogap and superconducting phases in magic-angle graphene.

Electron–electron interactions in magic-angle twisted bilayer graphene can split usually degenerate electronic bands, giving rise to a cascade of electronic transitions revealed by spectroscopy.

Insulating states that are formed because of pairing between electrons and holes are known to exist in engineered bilayer structures in high magnetic fields. Now evidence suggests they can occur in a monolayer crystal at zero field.

A magneto-transport study of twisted bilayer graphene near the magic angle further reveals its rich physics.

High-resolution scanning tunnelling microscopy measurements show that chains of magnetic atoms on the surface of a superconductor provide a promising platform for realizing and manipulating Majorana fermion quasiparticles.

Recent theoretical and experimental progress in identifying and understanding magnetic topological materials is reviewed, highlighting the antiferromagnetic topological insulator MnBi₂Te₄ and the ferromagnetic Weyl semimetal Co₃Sn₂S₂, and future research directions are discussed.