Collective modes in Fermi liquids are usually regarded as dissipation channels that relax electronic excitations through Landau damping. Whether such modes can instead mediate the formation of correlated electronic states under non-equilibrium conditions remains an open question. Here we show that, under optical photo-doping, a bulk plasmon can drive correlated inter-band transfer within a transient electronic continuum. Using time- and angle-resolved photoemission spectroscopy (Tr-ARPES) on EuCd_2As_2 supported by electronic structure calculations, we observe that at high excitation density, plasmons transfer energy from a weakly dispersing bulk band into unoccupied surface states. This bulk-to-surface redistribution stabilizes a long-lived, energy-localized spectral feature consistent with a Mahan exciton. Our results uncover a non-equilibrium regime of Fermi-liquid physics in which collective modes do not merely dissipate energy, but also stabilize correlated bound states.

Since the prediction and observation of topological Weyl semimetals (chiral TSMs), there have been enormous efforts to characterize further condensed matter realizations of chiral fermions. These efforts were dramatically accelerated by the subsequent discovery of a profound link between low-energy topological and lattice chirality in structurally chiral crystals. Though TSMs are well understood in the limit of perfect translation symmetry, real solid-state materials host defects and disorder, and may even be rendered amorphous down to all but the smallest system length scales. Previous theoretical studies have concluded that chiral TSMs transition into trivial diffusive metals at moderate disorder scales, raising concerns that chiral TSM states may only be accessible in highly crystalline samples. In this work, we in contrast identify large families of chiral TSMs that persist under strong structural disorder - even into the amorphous regime. We show that amorphous chiral TSM phases can in particular be stabilized by the presence of long-range order in the local structural chirality. We present extensive analytic and numerical calculations demonstrating the existence of both Weyl and higher-charge chiral fermions in amorphous metals whose topology and spin and orbital angular momentum textures are tunable via the interplay of average symmetry and geometry. To distinguish and generate new realizations of strongly disordered chiral fermions, we introduce an analytic approach grounded in symmetry group theory. We then introduce an amorphous Wilson loop numerical method to characterize chiral fermions with quantized Berry curvature fluxes in metals with 3D structural disorder. Our findings bridge the crystalline and strongly disordered regimes of chiral TSMs, and indicate a clear route towards engineering geometry-enforced topology in non-crystalline materials and metamaterials.

Abstract Symmetry-protected topological crystalline insulators (TCIs) have primarily been characterized by their gapless boundary states. However, in time-reversal- ( $${{{{{{{\mathcal{T}}}}}}}}$$ T -) invariant (helical) 3D TCIs—termed higher-order TCIs (HOTIs)—the boundary signatures can manifest as a sample-dependent network of 1D hinge states. We here introduce nested spin-resolved Wilson loops and layer constructions as tools to characterize the intrinsic bulk topological properties of spinful 3D insulators. We discover that helical HOTIs realize one of three spin-resolved phases with distinct responses that are quantitatively robust to large deformations of the bulk spin-orbital texture: 3D quantum spin Hall insulators (QSHIs), “spin-Weyl” semimetals, and $${{{{{{{\mathcal{T}}}}}}}}$$ T -doubled axion insulator (T-DAXI) states with nontrivial partial axion angles indicative of a 3D spin-magnetoelectric bulk response and half-quantized 2D TI surface states originating from a partial parity anomaly. Using ab-initio calculations, we demonstrate that β-MoTe2 realizes a spin-Weyl state and that α-BiBr hosts both 3D QSHI and T-DAXI regimes.

The group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry.However, the group theory of crystals with commensurate magnetic order has remained incomplete for the past 70 years, due to the complicated symmetries of magnetic crystals.In this work, we (i) complete the the representation theory of the magnetic space group by deriving the small corepresentations, momentum stars, compatibility relations, and magnetic elementary band corepresentations of the 1,421 magnetic space groups (MSGs).We also (ii) extend Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids -Magnetic Topological Quantum Chemistry (MTQC).Using a combination of real space layer construction and the MTQC method, we derive the complete set of symmetrybased indicators of electronic band topology, for which we identify symmetry-respecting bulk and anomalous surface and hinge states.In the end, we (iii) fully classify the spin space groups (SSGs) (more than 50 thousands) that characterize magnetic materials with negligible spin-orbit coupling and discuss exotic topological states protected by SSGs.

Topological insulating (TI) phases were originally highlighted for their disorder-robust bulk responses, such as the quantized Hall conductivity of 2D Chern insulators. With the discovery of time-reversal- ($\mathcal{T}$-) invariant 2D TIs, and the recognition that their spin Hall conductivity is generically non-quantized, focus has since shifted to boundary states as signatures of 2D and 3D TIs and symmetry-enforced topological crystalline insulators (TCIs). However, in $\mathcal{T}$-invariant (helical) 3D TCIs such as bismuth, $\alpha$-BiBr, and MoTe$_2$-termed higher-order TCIs (HOTIs)-the boundary signatures manifest as 1D hinge states, whose configurations are dependent on sample details, and bulk signatures remain unknown. In this work, we introduce nested spin-resolved Wilson loops and layer constructions as tools to characterize the bulk topological properties of spinful 3D insulators. We discover that helical HOTIs realize one of three spin-resolved phases with distinct responses that are quantitatively robust to large deformations of the bulk spin-orbital texture: 3D quantum spin Hall insulators (QSHIs),"spin-Weyl"semimetal states with gapless spin spectra, and $\mathcal{T}$-doubled axion insulator (T-DAXI) states with nontrivial partial axion angles $\theta^\pm = \pi$ indicative of a 3D spin-magnetoelectric bulk response. We provide experimental signatures of each spin-stable regime of helical HOTIs, including an extensive bulk spin Hall response in 3D QSHIs and half-quantized 2D TI states on the gapped surfaces of T-DAXIs originating from a partial parity anomaly. We use ab-initio calculations to compute the spin-resolved topology of candidate helical HOTIs, finding that $\beta$-MoTe$_2$ realizes a spin-Weyl state and that $\alpha$-BiBr hosts both 3D QSHI and T-DAXI regimes.

In insulating crystals, it was previously shown that defects with two fewer dimensions than the bulk can bind topological electronic states. We here further extend the classification of topological defect states by demonstrating that the corners of crystalline defects with integer Burgers vectors can bind 0D higher-order end (HEND) states with anomalous charge and spin. We demonstrate that HEND states are intrinsic topological consequences of the bulk electronic structure and introduce new bulk topological invariants that are predictive of HEND dislocation states in solid-state materials. We demonstrate the presence of first-order 0D defect states in PbTe monolayers and HEND states in 3D SnTe crystals. We relate our analysis to magnetic flux insertion in insulating crystals. We find that $π$-flux tubes in inversion- and time-reversal-symmetric (helical) higher-order topological insulators bind Kramers pairs of spin-charge-separated HEND states, which represent observable signatures of anomalous surface half quantum spin Hall states.

Topological quantum chemistry and symmetry-based indicators have facilitated large-scale searches for materials with topological properties at the Fermi energy (E-F). We report the implementation of a publicly accessible catalog of stable and fragile topology in all of the bands both at and away from E-F in the 96,196 processable entries in the Inorganic Crystal Structure Database. Our calculations, which represent the completion of the symmetry-indicated band topology of known nonmagnetic materials, have enabled the discovery of repeat-topological and supertopological materials, including rhombohedral bismuth and Bi2Mg3. We find that 52.65% of all materials are topological at E-F, roughly two-thirds of bands across all materials exhibit symmetry-indicated stable topology, and 87.99% of all materials contain at least one stable or fragile topological band.

A Correction to this paper has been published: https://doi.org/10.1038/s41567-021-01169-4 .

Charge density waves (CDWs) in Weyl semimetals (WSMs) have been shown to induce an exotic axionic insulating phase in which the sliding mode (phason) of the CDW acts as a dynamical axion field, giving rise to a large positive magnetoconductance [Wang et al., Phys. Rev. B 87, 161107(R) (2013); Roy et al., Phys. Rev. B 92, 125141 (2015); J. Gooth et al., Nature (London) 575, 315 (2019)]. In this work, we predict that dynamical strain can induce a bulk orbital magnetization in time-reversal (TR)-invariant WSMs that are gapped by a CDW. We term this effect the "dynamical piezomagnetic effect" (DPME). Unlike in J. Gooth et al. [Nature (London) 575, 315 (2019)], the DPME introduced in this work occurs in a bulk-constant (i.e., static and spatially homogeneous in the bulk) CDW, and does not rely on fluctuations, such as a phason. By studying the low-energy effective theory and a minimal tight-binding (TB) model, we find that the DPME originates from an effective valley axion field that couples the electromagnetic gauge field with a strain-induced pseudogauge field. In particular, whereas the piezoelectric effects studied in previous works are characterized by 2D Berry curvature, the DPME represents the first example of afundamentally 3D strain effect originating from the Chern-Simons 3-form. We further find that the DPME has a discontinuous change at a critical value of the phase of the CDW order parameter. We demonstrate that, when there is a jump in the DPME, the surface of the system undergoes a topological quantum phase transition (TQPT), while the bulk remains gapped. Hence, the DPME provides a bulk signature of the boundary TQPT in a TR-invariant Weyl CDW.

Topological materials discovery has evolved at a rapid pace over the past 15 years following the identification of the first nonmagnetic topological insulators (TIs), topological crystalline insulators (TCIs) and 3D topological semimetals (TSMs). Most recently, through complete analyses of symmetry-allowed band structures — including the theory of topological quantum chemistry (TQC) — researchers have determined crystal-symmetry-enhanced Wilson-loop and complete symmetry-based indicators for nonmagnetic topological phases, leading to the discovery of higher-order TCIs and TSMs. The recent application of TQC and related methods to high-throughput materials discovery has revealed that over half of the known stoichiometric, solid-state, nonmagnetic materials are topological at the Fermi level, over 85 per cent of the known stoichiometric materials host energetically isolated topological bands, and just under two-thirds of the energetically isolated bands in known materials carry the stable topology of a TI or TCI. In this Review, we survey topological electronic materials discovery in nonmagnetic crystalline solids from the prediction of the first 2D and 3D TIs to the recently introduced methods that have facilitated large-scale searches for topological materials. We also discuss future venues for the identification and manipulation of solid-state topological phases, including charge-density-wave compounds, magnetic materials, and 2D few-layer devices. 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.

A Correction to this paper has been published: .

Topological physics and strong electron–electron correlations in quantum materials are typically studied independently. However, there have been rapid recent developments in quantum materials in which topological phase transitions emerge when the single-particle band structure is modified by strong interactions. Here we demonstrate that the room-temperature phase of (TaSe 4 ) 2 I is a Weyl semimetal with 24 pairs of Weyl nodes. Owing to its quasi-one-dimensional structure, (TaSe 4 ) 2 I also hosts an established charge-density wave instability just below room temperature. We show that the charge-density wave in (TaSe 4 ) 2 I couples the bulk Weyl points and opens a bandgap. The correlation-driven topological phase transition in (TaSe 4 ) 2 I provides a route towards observing condensed-matter realizations of axion electrodynamics in the gapped regime, topological chiral response effects in the semimetallic phase, and represents an avenue for exploring the interplay of correlations and topology in a solid-state material. Strong electron–electron interactions create a charge-density wave that modifies the topological state of the Weyl semimetal (TaSe 4 ) 2 I. This implies the possibility of experimentally simulating axion electrodynamics in a solid-state material.

For over 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. However, the group theory of crystals with commensurate magnetic order has remained incomplete for the past 70 years, due to the complicated symmetries of magnetic crystals. In this work, we complete the 100-year-old problem of crystalline group theory by deriving the small corepresentations, momentum stars, compatibility relations, and magnetic elementary band corepresentations of the 1,421 magnetic space groups (MSGs), which we have made freely accessible through tools on the Bilbao Crystallographic Server. We extend Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids - Magnetic Topological Quantum Chemistry (MTQC). Using MTQC, we derive the complete set of symmetry-based indicators of electronic band topology, for which we identify symmetry-respecting bulk and anomalous surface and hinge states.

In recent theoretical and experimental investigations, researchers have linked the low-energy field theory of a Weyl semimetal gapped with a charge-density wave (CDW) to high-energy theories with axion electrodynamics. However, it remains an open question whether a lattice regularization of the dynamical Weyl-CDW is in fact a single-particle axion insulator (AXI). In this Rapid Communication, we use analytic and numerical methods to study both lattice-commensurate and incommensurate minimal (magnetic) Weyl-CDW phases in the mean-field state. We observe that, as previously predicted from field theory, the two inversion (I)-symmetric Weyl-CDWs with phi = 0, pi differ by a topological axion angle delta theta(phi) = pi. However, we crucially discover that neither of the minimalWeyl-CDWphases at phi = 0, pi is individually an AXI; they are instead quantum anomalous Hall (QAH) and "obstructed" QAH insulators that differ by a fractional translation in the modulated cell, analogous to the two phases of the Su-Schrieffer-Heeger model of polyacetylene. Using symmetry indicators of band topology and non-Abelian Berry phase, we demonstrate that our results generalize to multiband systems with only two Weyl fermions, establishing that minimal Weyl-CDWs unavoidably carry nontrivial Chern numbers that prevent the observation of a static magnetoelectric response. We discuss the experimental implications of our findings and provide models and analysis generalizing our results to nonmagnetic Weyl- and Dirac-CDWs.