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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.

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.

Whereas ferromagnets have been known and used for millennia, antiferromagnets were only discovered in the 1930s.sup.1. At large scale, because of the absence of global magnetization, antiferromagnets may seem to behave like any non-magnetic material. At the microscopic level, however, the opposite alignment of spins forms a rich internal structure. In topological antiferromagnets, this internal structure leads to the possibility that the property known as the Berry phase can acquire distinct spatial textures.sup.2,3. Here we study this possibility in an antiferromagnetic axion insulator--even-layered, two-dimensional MnBi.sub.2Te.sub.4--in which spatial degrees of freedom correspond to different layers. We observe a type of Hall effect--the layer Hall effect--in which electrons from the top and bottom layers spontaneously deflect in opposite directions. Specifically, under zero electric field, even-layered MnBi.sub.2Te.sub.4 shows no anomalous Hall effect. However, applying an electric field leads to the emergence of a large, layer-polarized anomalous Hall effect of about 0.5e.sup.2/h (where e is the electron charge and h is Planck's constant). This layer Hall effect uncovers an unusual layer-locked Berry curvature, which serves to characterize the axion insulator state. Moreover, we find that the layer-locked Berry curvature can be manipulated by the axion field formed from the dot product of the electric and magnetic field vectors. Our results offer new pathways to detect and manipulate the internal spatial structure of fully compensated topological antiferromagnets.sup.4-9. The layer-locked Berry curvature represents a first step towards spatial engineering of the Berry phase through effects such as layer-specific moiré potential. A new type of Hall effect--the layer Hall effect--is produced in a 2D antiferromagnet that does not exhibit any net magnetization., Author(s): Anyuan Gao [sup.1] , Yu-Fei Liu [sup.1] , Chaowei Hu [sup.2] , Jian-Xiang Qiu [sup.1] , Christian Tzschaschel [sup.1] , Barun Ghosh [sup.3] [sup.4] , Sheng-Chin Ho [sup.1] , [...]

Using a recently developed formalism called topological quantum chemistry, we perform a high-throughput search of 'high-quality' materials (for which the atomic positions and structure have been measured very accurately) in the Inorganic Crystal Structure Database in order to identify new topological phases. We develop codes to compute all characters of all symmetries of 26,938 stoichiometric materials, and find 3,307 topological insulators, 4,078 topological semimetals and no fragile phases. For these 7,385 materials we provide the electronic band structure, including some electronic properties (bandgap and number of electrons), symmetry indicators, and other topological information. Our results show that more than 27 per cent of all materials in nature are topological. We provide an open-source code that checks the topology of any material and allows other researchers to reproduce our results. 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., Author(s): M. G. Vergniory [sup.1] [sup.2] [sup.3] , L. Elcoro [sup.4] , Claudia Felser [sup.5] , Nicolas Regnault [sup.6] , B. Andrei Bernevig [sup.7] [sup.8] [sup.9] , Zhijun Wang [sup.7] [...]

Author(s): Johannes Gooth (corresponding author) [1, 2]; Anna C. Niemann [1, 3]; Tobias Meng [4]; Adolfo G. Grushin [5]; Karl Landsteiner [6]; Bernd Gotsmann [2]; Fabian Menges [2]; Marcus Schmidt [...]

When electric conductors differ from their mirror image, unusual chiral transport coefficients appear that are forbidden in achiral metals, such as a non-linear electric response known as electronic magnetochiral anisotropy (eMChA)1–6. Although chiral transport signatures are allowed by symmetry in many conductors without a centre of inversion, they reach appreciable levels only in rare cases in which an exceptionally strong chiral coupling to the itinerant electrons is present. So far, observations of chiral transport have been limited to materials in which the atomic positions strongly break mirror symmetries. Here, we report chiral transport in the centrosymmetric layered kagome metal CsV3Sb5 observed via second-harmonic generation under an in-plane magnetic field. The eMChA signal becomes significant only at temperatures below T ′ ≈ 35 K, deep within the charge-ordered state of CsV3Sb5 (TCDW ≈ 94 K). This temperature dependence reveals a direct correspondence between electronic chirality, unidirectional charge order7 and spontaneous time-reversal symmetry breaking due to putative orbital loop currents8–10. We show that the chirality is set by the out-of-plane field component and that a transition from left- to right-handed transport can be induced by changing the field sign. CsV3Sb5 is the first material in which strong chiral transport can be controlled and switched by small magnetic field changes, in stark contrast to structurally chiral materials, which is a prerequisite for applications in chiral electronics.Change of chirality from left- to right-handed transport in the layered kagome metal CsV3Sb5 can be controlled by small magnetic field changes, a required feature for chiral electronic applications. [ABSTRACT FROM AUTHOR]

Author(s): Ajaya K. Nayak [1]; Vivek Kumar [2]; Tianping Ma [1]; Peter Werner [1]; Eckhard Pippel [1]; Roshnee Sahoo [2]; Franoise Damay [3]; Ulrich K. Rler [4]; Claudia Felser [2]; [...]

An amendment to this paper has been published and can be accessed via a link at the top of the paper., Author(s): M. G. Vergniory [sup.1] [sup.2] [sup.3] , L. Elcoro [sup.4] , Claudia Felser [sup.5] , Nicolas Regnault [sup.6] , B. Andrei Bernevig [sup.7] [sup.8] [sup.9] , Zhijun Wang [sup.7] [...]