Observation of a linked-loop quantum state in a topological magnet.

Quantum phases can be classified by topological invariants, which take on discrete values capturing global information about the quantum state1-13. Over the past decades, these invariants have come to play a central role in describing matter, providing the foundation for understanding superfluids5, magnets6,7, the quantum Hall effect3,8, topological insulators9,10, Weyl semimetals11-13 and other phenomena. Here we report an unusual linking-number (knot theory) invariant associated with loops of electronic band crossings in a mirror-symmetric ferromagnet14-20. Using state-of-the-art spectroscopic methods, we directly observe three intertwined degeneracy loops in the material's three-torus, T3, bulk Brillouin zone. We find that each loop links each other loop twice. Through systematic spectroscopic investigation of this linked-loop quantum state, we explicitly draw its link diagram and conclude, in analogy with knot theory, that it exhibits the linking number (2, 2, 2), providing a direct determination of the invariant structure from the experimental data. We further predict and observe, on the surface of our samples, Seifert boundary states protected by the bulk linked loops, suggestive of a remarkable Seifert bulk-boundary correspondence. Our observation of a quantum loop link motivates the application of knot theory to the exploration of magnetic and superconducting quantum matter.

Topological chiral crystals with helicoid-arc quantum states.

The quantum behaviour of electrons in materials is the foundation of modern electronics and information technology1-11, and quantum materials with topological electronic and optical properties are essential for realizing quantized electronic responses that can be used for next generation technology. Here we report the first observation of topological quantum properties of chiral crystals6,7 in the RhSi family. We find that this material class hosts a quantum phase of matter that exhibits nearly ideal topological surface properties originating from the crystals' structural chirality. Electrons on the surface of these crystals show a highly unusual helicoid fermionic structure that spirals around two high-symmetry momenta, indicating electronic topological chirality. The existence of bulk multiply degenerate band fermions is guaranteed by the crystal symmetries; however, to determine the topological invariant or charge in these chiral crystals, it is essential to identify and study the helicoid topology of the arc states. The helicoid arcs that we observe on the surface characterize the topological charges of ±2, which arise from bulk higher-spin chiral fermions. These topological conductors exhibit giant Fermi arcs of maximum length (π), which are orders of magnitude larger than those found in known chiral Weyl fermion semimetals5,8-11. Our results demonstrate an electronic topological state of matter on structurally chiral crystals featuring helicoid-arc quantum states. Such exotic multifold chiral fermion semimetal states could be used to detect a quantized photogalvanic optical response, the chiral magnetic effect and other optoelectronic phenomena predicted for this class of materials6.

Visualizing Higher-Fold Topology in Chiral Crystals.

Novel topological phases of matter are fruitful platforms for the discovery of unconventional electromagnetic phenomena. Higher-fold topology is one example, where the low-energy description goes beyond standard model analogs. Despite intensive experimental studies, conclusive evidence remains elusive for the multigap topological nature of higher-fold chiral fermions. In this Letter, we leverage a combination of fine-tuned chemical engineering and photoemission spectroscopy with photon energy contrast to discover the higher-fold topology of a chiral crystal. We identify all bulk branches of a higher-fold chiral fermion for the first time, critically important for allowing us to explore unique Fermi arc surface states in multiple interband gaps, which exhibit an emergent ladder structure. Through designer chemical gating of the samples in combination with our measurements, we uncover an unprecedented multigap bulk boundary correspondence. Our demonstration of multigap electronic topology will propel future research on unconventional topological responses.

Topological Quantum Materials from the Viewpoint of Chemistry.

Topology, a mathematical concept, has recently become a popular and truly transdisciplinary topic encompassing condensed matter physics, solid state chemistry, and materials science. Since there is a direct connection between real space, namely atoms, valence electrons, bonds, and orbitals, and reciprocal space, namely bands and Fermi surfaces, via symmetry and topology, classifying topological materials within a single-particle picture is possible. Currently, most materials are classified as trivial insulators, semimetals, and metals or as topological insulators, Dirac and Weyl nodal-line semimetals, and topological metals. The key ingredients for topology are certain symmetries, the inert pair effect of the outer electrons leading to inversion of the conduction and valence bands, and spin-orbit coupling. This review presents the topological concepts related to solids from the viewpoint of a solid-state chemist, summarizes techniques for growing single crystals, and describes basic physical property measurement techniques to characterize topological materials beyond their structure and provide examples of such materials. Finally, a brief outlook on the impact of topology in other areas of chemistry is provided at the end of the article.

Optical signatures of multifold fermions in the chiral topological semimetal CoSi.

We report the optical conductivity in high-quality crystals of the chiral topological semimetal CoSi, which hosts exotic quasiparticles known as multifold fermions. We find that the optical response is separated into several distinct regions as a function of frequency, each dominated by different types of quasiparticles. The low-frequency intraband response is captured by a narrow Drude peak from a high-mobility electron pocket of double Weyl quasiparticles, and the temperature dependence of the spectral weight is consistent with its Fermi velocity. By subtracting the low-frequency sharp Drude and phonon peaks at low temperatures, we reveal two intermediate quasilinear interband contributions separated by a kink at 0.2 eV. Using Wannier tight-binding models based on first-principle calculations, we link the optical conductivity above and below 0.2 eV to interband transitions near the double Weyl fermion and a threefold fermion, respectively. We analyze and determine the chemical potential relative to the energy of the threefold fermion, revealing the importance of transitions between a linearly dispersing band and a flat band. More strikingly, below 0.1 eV our data are best explained if spin-orbit coupling is included, suggesting that at these energies, the optical response is governed by transitions between a previously unobserved fourfold spin-3/2 node and a Weyl node. Our comprehensive combined experimental and theoretical study provides a way to resolve different types of multifold fermions in CoSi at different energy. More broadly, our results provide the necessary basis to interpret the burgeoning set of optical and transport experiments in chiral topological semimetals.

Observation of Asymmetric Oxidation Catalysis with B20 Chiral Crystals.

The study of heterogeneous reactions for enantiomeric processes based on inorganic crystals has been resurgent in recent years. However, the question remains how homochirality develops in nature and chemical reactions. Here, the successful growth of B20 group PdGa single crystals with different chiral lattices enabled us to achieve enantioselective recognition of 3,4-dihydroxyphenylalanine (DOPA) based on a new mechanism, namely orbital angular momentum (OAM) polarization. The orbital textures of PdGa crystals indicate large OAM polarization near the Fermi level and carrying opposite signs. A positive or negative magnetization in the [111] direction is expected depending on the chiral lattice of PdGa crystals. Due to this, the adsorption energies of PdGa crystals and DOPA molecules differ depending on how well the O-2p orbital of DOPA pairs with the Pd-4d orbital of PdGa. The results provide one possible explanation for how chirality arises in nature by providing an enantioselective route with pure inorganic crystals.

Direct Measurement of Helicoid Surface States in RhSi Using Nonlinear Optics.

Despite the fundamental nature of the edge state in topological physics, direct measurement of electronic and optical properties of the Fermi arcs of topological semimetals has posed a significant experimental challenge, as their response is often overwhelmed by the metallic bulk. However, laser-driven currents carried by surface and bulk states can propagate in different directions in nonsymmorphic crystals, allowing for the two components to be easily separated. Motivated by a recent theoretical prediction G. Chang et al., Phys. Rev. Lett. 124, 166404 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.166404, we have measured the linear and circular photogalvanic effect currents deriving from the Fermi arcs of the nonsymmorphic, chiral Weyl semimetal RhSi over the 0.45-1.1 eV incident photon energy range. Our data are in good agreement with the predicted spectral shape of the circular photogalvanic effect as a function of photon energy, although the direction of the surface photocurrent departed from the theoretical expectation over the energy range studied. Surface currents arising from the linear photogalvanic effect were observed as well, with the unexpected result that only two of the six allowed tensor element were required to describe the measurements, suggesting an approximate emergent mirror symmetry inconsistent with the space group of the crystal.

Tunable eg Orbital Occupancy in Heusler Compounds for Oxygen Evolution Reaction*.

Heusler compounds have potential in electrocatalysis because of their mechanical robustness, metallic conductivity, and wide tunability in the electronic structure and element compositions. This study reports the first application of Co2 YZ-type Heusler compounds as electrocatalysts for the oxygen evolution reaction (OER). A range of Co2 YZ crystals was synthesized through the arc-melting method and the eg orbital filling of Co was precisely regulated by varying Y and Z sites of the compound. A correlation between the eg orbital filling of reactive Co sites and OER activity was found for Co2 MnZ compounds (Z=Ti, Al, V, and Ga), whereby higher catalytic current was achieved for eg orbital filling approaching unity. A similar trend of eg orbital filling on the reactivity of cobalt sites was also observed for other Heusler compounds (Co2 VZ, Z=Sn and Ga). This work demonstrates proof of concept in the application of Heusler compounds as a new class of OER electrocatalysts, and the influence of the manipulation of the spin orbitals on their catalytic performance.

Low thermal conductivity in a new mixed metal telluride Mn1.8(1)In0.8(1)Si2Te6.

The design of new complex mixed metal tellurides (containing low toxicity cations) with intrinsic ultralow thermal conductivity is of paramount importance in the field of thermoelectrics. Herein, we report the synthesis and characterization of polycrystalline and single crystals of a new mixed-metal quaternary telluride Mn1.8(1)In0.8(1)Si2Te6. The structural aspects and chemical formula of this phase at room temperature have been established using single crystal X-ray diffraction and EDX studies. The trigonal centrosymmetric (space group: P3̄1m) structure of the title phase has cell constants of a = b = 7.0483(7) Å and c = 7.1277(8) Å. The structure has three independent cationic sites, one mixed (In1/Mn1), one partially filled Mn2, and one Si1, which are bonded with Te1 atoms. Each metal atom (In and Mn) in the structure is octahedrally coordinated with six neighboring Te1 atoms. The structure also features dimers of Si atoms, and each Si atom is bonded to three Te1 atoms to form ethane-like Si2Te6 units. The optical absorption study of a polycrystalline Mn1.8In0.8Si2Te6 sample shows a narrow optical bandgap of 0.6(2) eV. Temperature-dependent resistivity and Seebeck coefficient studies confirmed the p-type semiconducting nature of the sample with high values of S (301 µV K-1 to 444 µV K-1). The total thermal conductivity (ktot) study of the polycrystalline sample shows a decreasing trend on heating with an extremely low value of 0.28 W m-1 K-1 at 773 K. Magnetic measurements indicate a glassy magnetic behavior for the sample below 8 K.

Weyl spin-momentum locking in a chiral topological semimetal.

Spin-orbit coupling in noncentrosymmetric crystals leads to spin-momentum locking - a directional relationship between an electron's spin angular momentum and its linear momentum. Isotropic orthogonal Rashba spin-momentum locking has been studied for decades, while its counterpart, isotropic parallel Weyl spin-momentum locking has remained elusive in experiments. Theory predicts that Weyl spin-momentum locking can only be realized in structurally chiral cubic crystals in the vicinity of Kramers-Weyl or multifold fermions. Here, we use spin- and angle-resolved photoemission spectroscopy to evidence Weyl spin-momentum locking of multifold fermions in the chiral topological semimetal PtGa. We find that the electron spin of the Fermi arc surface states is orthogonal to their Fermi surface contour for momenta close to the projection of the bulk multifold fermion at the Γ point, which is consistent with Weyl spin-momentum locking of the latter. The direct measurement of the bulk spin texture of the multifold fermion at the R point also displays Weyl spin-momentum locking. The discovery of Weyl spin-momentum locking may lead to energy-efficient memory devices and Josephson diodes based on chiral topological semimetals.

Anisotropic resistance with a 90° twist in a ferromagnetic Weyl semimetal, Co2MnGa.

Weyl semimetals exhibit exotic magnetotransport phenomena such as the chiral anomaly and surface-to-bulk quantum oscillations (Weyl orbits) due to chiral bulk states and topologically protected surface states. Here we report a unique transport property in crystals of the ferromagnetic nodal-line Weyl semimetal Co2MnGa that have been polished to micron thicknesses using a focused ion beam. These thin crystals exhibit a large planar resistance anisotropy (10 × ) with axes that rotate by 90 degrees between opposite faces of the crystal. We use symmetry arguments and electrostatic simulations to show that the observed anisotropy resembles that of an isotropic conductor with surface states that are impeded from hybridization with bulk states. The origin of these states awaits further experiments that can correlate the surface bands with the observed 90° twist.

Observation of a phase transition within the domain walls of ferromagnetic Co3Sn2S2.

The ferromagnetic phase of Co3Sn2S2 is widely considered to be a topological Weyl semimetal, with evidence for momentum-space monopoles of Berry curvature from transport and spectroscopic probes. As the bandstructure is highly sensitive to the magnetic order, attention has focused on anomalies in magnetization, susceptibility and transport measurements that are seen well below the Curie temperature, leading to speculation that a "hidden" phase coexists with ferromagnetism. Here we report spatially-resolved measurements by Kerr effect microscopy that identify this phase. We find that the anomalies coincide with a deep minimum in domain wall (DW) mobility, indicating a crossover between two regimes of DW propagation. We demonstrate that this crossover is a manifestation of a 2D phase transition that occurs within the DW, in which the magnetization texture changes from continuous rotation to unidirectional variation. We propose that the existence of this 2D transition deep within the ferromagnetic state of the bulk is a consequence of a giant quality factor for magnetocrystalline anisotropy unique to this compound. This work broadens the horizon of the conventional binary classification of DWs into Bloch and Néel walls, and suggests new strategies for manipulation of domain walls and their role in electron and spin transport.

Quasi-symmetry protected topology in a semi-metal.

The crystal symmetry of a material dictates the type of topological band structures it may host, and therefore symmetry is the guiding principle to find topological materials. Here we introduce an alternative guiding principle, which we call 'quasi-symmetry'. This is the situation where a Hamiltonian has an exact symmetry at lower-order that is broken by higher-order perturbation terms. This enforces finite but parametrically small gaps at some low-symmetry points in momentum space. Untethered from the restraints of symmetry, quasi-symmetries eliminate the need for fine-tuning as they enforce that sources of large Berry curvature will occur at arbitrary chemical potentials. We demonstrate that a quasi-symmetry in the semi-metal CoSi stabilizes gaps below 2 meV over a large near-degenerate plane that can be measured in the quantum oscillation spectrum. The application of in-plane strain breaks the crystal symmetry and gaps the degenerate point, observable by new magnetic breakdown orbits. The quasi-symmetry, however, does not depend on spatial symmetries and hence transmission remains fully coherent. These results demonstrate a class of topological materials with increased resilience to perturbations such as strain-induced crystalline symmetry breaking, which may lead to robust topological applications as well as unexpected topology beyond the usual space group classifications.

Large Anomalous Hall and Nernst Effects in High Curie-Temperature Iron-Based Heusler Compounds.

The interplay between topology and magnetism has recently sparked the frontier studies of magnetic topological materials that exhibit intriguing anomalous Hall and Nernst effects owning to the large intrinsic Berry curvature (BC). To better understand the anomalous quantum transport properties of these materials and their implications for future applications such as electronic and thermoelectric devices, it is crucial to discover more novel material platforms for performing anomalous transverse transport studies. Here, it is experimentally demonstrated that low-cost Fe-based Heusler compounds exhibit large anomalous Hall and Nernst effects. An anomalous Hall conductivity of 250-750 S cm-1 and Nernst thermopower of above 2 µV K-1 are observed near room temperature. The positive effect of anti-site disorder on the anomalous Hall transport is revealed. Considering the very high Curie temperature (nearly 1000 K), larger Nernst thermopowers at high temperatures are expected owing to the existing magnetic order and the intrinsic BC. This work provides a background for developing low-cost Fe-based Heusler compounds as a new material platform for anomalous transport studies and applications, in particular, near and above room temperature.

2D-Berry-Curvature-Driven Large Anomalous Hall Effect in Layered Topological Nodal-Line MnAlGe.

Topological magnets comprising 2D magnetic layers with Curie temperatures (TC ) exceeding room temperature are key for dissipationless quantum transport devices. However, the identification of a material with 2D ferromagnetic planes that exhibits an out-of-plane-magnetization remains a challenge. This study reports a ferromagnetic, topological, nodal-line, and semimetal MnAlGe composed of square-net Mn layers that are separated by nonmagnetic Al-Ge spacers. The 2D ferromagnetic Mn layers exhibit an out-of-plane magnetization below TC  ≈ 503 K. Density functional calculations demonstrate that 2D arrays of Mn atoms control the electrical, magnetic, and therefore topological properties in MnAlGe. The unique 2D distribution of the Berry curvature resembles the 2D Fermi surface of the bands that form the topological nodal line near the Fermi energy. A large anomalous Hall conductivity of ≈700 S cm-1 is obtained at 2 K and related to this nodal-line-induced 2D Berry curvature distribution. The high transition temperature, large anisotropic out-of-plane magnetism, and natural heterostructure-type atomic arrangements consisting of magnetic Mn and nonmagnetic Al/Ge elements render nodal-line MnAlGe one of the few, unique, and layered topological ferromagnets that have ever been observed.

Topological Engineering of Pt-Group-Metal-Based Chiral Crystals toward High-Efficiency Hydrogen Evolution Catalysts.

It has been demonstrated that topological nontrivial surface states can favor heterogeneous catalysis processes such as the hydrogen evolution reaction (HER), but a further decrease in mass loading and an increase in activity are still highly challenging. The observation of massless chiral fermions associated with large topological charge and long Fermi arc (FA) surface states inspires the investigation of their relationship with the charge transfer and adsorption process in the HER. In this study, it is found that the HER efficiency of Pt-group metals can be boosted significantly by introducing topological order. A giant nontrivial topological energy window and a long topological surface FA are expected at the surface when forming chiral crystals in the space group of P21 3 (#198). This makes the nontrivial topological features resistant to a large change in the applied overpotential. As HER catalysts, PtAl and PtGa chiral crystals show turnover frequencies as high as 5.6 and 17.1 s-1 and an overpotential as low as 14 and 13.3 mV at a current density of 10 mA cm-2 . These crystals outperform those of commercial Pt and nanostructured catalysts. This work opens a new avenue for the development of high-efficiency catalysts with the strategy of topological engineering of excellent transitional catalytic materials.

Helicity-dependent photocurrents in the chiral Weyl semimetal RhSi.

Weyl semimetals are crystals in which electron bands cross at isolated points in momentum space. Associated with each crossing point (or Weyl node) is a topological invariant known as the Berry monopole charge. The circular photogalvanic effect (CPGE), whereby circular polarized light generates a helicity-dependent photocurrent, is a notable example of a macroscopic property that emerges directly from the topology of the Weyl semimetal band structure. Recently, it was predicted that the amplitude of the CPGE associated with optical transitions near a Weyl node is proportional to its monopole charge. In chiral Weyl systems, nodes of opposite charge are nondegenerate, opening a window of wavelengths where the CPGE resulting from uncompensated Berry charge can emerge. Here, we report measurements of CPGE in the chiral Weyl semimetal RhSi, revealing a CPGE response in an energy window that closes at 0.65 eV, in agreement with the predictions of density functional theory.

Observation and control of maximal Chern numbers in a chiral topological semimetal.

Topological semimetals feature protected nodal band degeneracies characterized by a topological invariant known as the Chern number (C). Nodal band crossings with linear dispersion are expected to have at most [Formula: see text], which sets an upper limit to the magnitude of many topological phenomena in these materials. Here, we show that the chiral crystal palladium gallium (PdGa) displays multifold band crossings, which are connected by exactly four surface Fermi arcs, thus proving that they carry the maximal Chern number magnitude of 4. By comparing two enantiomers, we observe a reversal of their Fermi-arc velocities, which demonstrates that the handedness of chiral crystals can be used to control the sign of their Chern numbers.

Observation of giant spin-split Fermi-arc with maximal Chern number in the chiral topological semimetal PtGa.

Non-symmorphic chiral topological crystals host exotic multifold fermions, and their associated Fermi arcs helically wrap around and expand throughout the Brillouin zone between the high-symmetry center and surface-corner momenta. However, Fermi-arc splitting and realization of the theoretically proposed maximal Chern number rely heavily on the spin-orbit coupling (SOC) strength. In the present work, we investigate the topological states of a new chiral crystal, PtGa, which has the strongest SOC among all chiral crystals reported to date. With a comprehensive investigation using high-resolution angle-resolved photoemission spectroscopy, quantum-oscillation measurements, and state-of-the-art ab initio calculations, we report a giant SOC-induced splitting of both Fermi arcs and bulk states. Consequently, this study experimentally confirms the realization of a maximal Chern number equal to ±4 in multifold fermionic systems, thereby providing a platform to observe large-quantized photogalvanic currents in optical experiments.

A New Highly Anisotropic Rh-Based Heusler Compound for Magnetic Recording.

The development of high-density magnetic recording media is limited by superparamagnetism in very small ferromagnetic crystals. Hard magnetic materials with strong perpendicular anisotropy offer stability and high recording density. To overcome the difficulty of writing media with a large coercivity, heat-assisted magnetic recording was developed, rapidly heating the media to the Curie temperature Tc before writing, followed by rapid cooling. Requirements are a suitable Tc , coupled with anisotropic thermal conductivity and hard magnetic properties. Here, Rh2 CoSb is introduced as a new hard magnet with potential for thin-film magnetic recording. A magnetocrystalline anisotropy of 3.6 MJ m-3 is combined with a saturation magnetization of µ0 Ms  = 0.52 T at 2 K (2.2 MJ m-3 and 0.44 T at room temperature). The magnetic hardness parameter of 3.7 at room temperature is the highest observed for any rare-earth-free hard magnet. The anisotropy is related to an unquenched orbital moment of 0.42 µB on Co, which is hybridized with neighboring Rh atoms with a large spin-orbit interaction. Moreover, the pronounced temperature dependence of the anisotropy that follows from its Tc of 450 K, together with a thermal conductivity of 20 W m-1 K-1 , make Rh2 CoSb a candidate for the development of heat-assisted writing with a recording density in excess of 10 Tb in.-2 .