Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet.

Despite recent efforts to advance spintronics devices and quantum information technology using materials with non-trivial topological properties, three key challenges are still unresolved1-9. First, the identification of topological band degeneracies that are generically rather than accidentally located at the Fermi level. Second, the ability to easily control such topological degeneracies. And third, the identification of generic topological degeneracies in large, multisheeted Fermi surfaces. By combining de Haas-van Alphen spectroscopy with density functional theory and band-topology calculations, here we show that the non-symmorphic symmetries10-17 in chiral, ferromagnetic manganese silicide (MnSi) generate nodal planes (NPs)11,12, which enforce topological protectorates (TPs) with substantial Berry curvatures at the intersection of the NPs with the Fermi surface (FS) regardless of the complexity of the FS. We predict that these TPs will be accompanied by sizeable Fermi arcs subject to the direction of the magnetization. Deriving the symmetry conditions underlying topological NPs, we show that the 1,651 magnetic space groups comprise 7 grey groups and 26 black-and-white groups with topological NPs, including the space group of ferromagnetic MnSi. Thus, the identification of symmetry-enforced TPs, which can be controlled with a magnetic field, on the FS of MnSi suggests the existence of similar properties-amenable for technological exploitation-in a large number of materials.

Caging-Pnictogen-Induced Superconductivity in Skutterudites IrX3 (X = As, P).

Here, we report on a new kind of compound, XδIr4X12-δ (X = P, As), the first hole-doped skutterudites superconductor. We provide atomic-resolution images of the caging As atoms using scanning transmission electron microscopy (STEM). By inserting As atoms into the caged structure under a high pressure, superconductivity emerges with a maximum transition temperature (Tc) of 4.4 K (4.8 K) in IrAs3 (IrP3). In contrast to all of the electron-doped skutterudites, the electronic states around the Fermi level in XδIr4X12-δ are dominated by the caged X atom, which can be described by a simple body-centered tight-binding model, implying a distinct pairing mechanism. Our density functional theory (DFT) calculations reveal an intimate relationship between the pressure-dependent local-phonon mode and the enhancement of Tc. The discovery of XδIr4X12-δ provides an arena to investigate the uncharted territory of hole-doped skutterudites, and the method proposed here represents a new strategy of carrier doping in caged structures, without introducing extra elements.

Network of Topological Nodal Planes, Multifold Degeneracies, and Weyl Points in CoSi.

We showcase the importance of global band topology in a study of the Weyl semimetal CoSi as a representative of chiral space group (SG) 198. We identify a network of band crossings comprising topological nodal planes, multifold degeneracies, and Weyl points consistent with the fermion doubling theorem. To confirm these findings, we combined the general analysis of the band topology of SG 198 with Shubnikov-de Haas oscillations and material-specific calculations of the electronic structure and Berry curvature. The observation of two nearly dispersionless Shubnikov-de Haas frequency branches provides unambiguous evidence of four Fermi surface sheets at the R point that reflect the symmetry-enforced orthogonality of the underlying wave functions at the intersections with the nodal planes. Hence, irrespective of the spin-orbit coupling strength, SG 198 features always six- and fourfold degenerate crossings at R and Γ that are intimately connected to the topological charges distributed across the network.

Competition of Superconductivity and Charge Density Wave in Selective Oxidized CsV_{3}Sb_{5} Thin Flakes.

The recently discovered layered kagome metals AV_{3}Sb_{5} (A=K, Rb, and Cs) with vanadium kagome networks provide a novel platform to explore correlated quantum states intertwined with topological band structures. Here we report the prominent effect of hole doping on both superconductivity and charge density wave (CDW) order, achieved by selective oxidation of exfoliated thin flakes. A superconducting dome is revealed as a function of the effective doping content. The superconducting transition temperature (T_{c}) and upper critical field in thin flakes are significantly enhanced compared with the bulk, which are accompanied by the suppression of CDW. Our detailed analyses establish the pivotal role of van Hove singularities in promoting correlated quantum orders in these kagome metals. Our experiments not only demonstrate the intriguing nature of superconducting and CDW orders, but also provide a novel route to tune the carrier concentration through both selective oxidation and electric gating. This establishes CsV_{3}Sb_{5} as a tunable 2D platform for the further exploration of topology and correlation among 3d electrons in kagome lattices.

Nature of Unconventional Pairing in the Kagome Superconductors AV_{3}Sb_{5} (A=K,Rb,Cs).

The recent discovery of AV_{3}Sb_{5} (A=K,Rb,Cs) has uncovered an intriguing arena for exotic Fermi surface instabilities in a kagome metal. Among them, superconductivity is found in the vicinity of multiple van Hove singularities, exhibiting indications of unconventional pairing. We show that the sublattice interference mechanism is central to understanding the formation of superconductivity in a kagome metal. Starting from an appropriately chosen minimal tight-binding model with multiple van Hove singularities close to the Fermi level for AV_{3}Sb_{5}, we provide a random phase approximation analysis of superconducting instabilities. Nonlocal Coulomb repulsion, the sublattice profile of the van Hove bands, and the interaction strength turn out to be the crucial parameters to determine the preferred pairing symmetry. Implications for potentially topological surface states are discussed, along with a proposal for additional measurements to pin down the nature of superconductivity in AV_{3}Sb_{5}.

Mechanism for Unconventional Superconductivity in the Hole-Doped Rashba-Hubbard Model.

Motivated by the recent resurgence of interest in topological superconductivity, we study superconducting pairing instabilities of the hole-doped Rashba-Hubbard model on the square lattice with first- and second-neighbor hopping. Within the random phase approximation, we compute the spin-fluctuation-mediated pairing interactions as a function of filling. Rashba spin-orbit coupling splits the spin degeneracies of the bands, which leads to two van Hove singularities at two different fillings. We find that, for a doping region in between these two van Hove fillings, the spin fluctuations exhibit a strong ferromagnetic contribution. Because of these ferromagnetic fluctuations, there is a strong tendency towards spin-triplet f-wave pairing within this filling region, resulting in a topologically nontrivial phase.

Unified Theory of PT and CP Invariant Topological Metals and Nodal Superconductors.

As PT and CP symmetries are fundamental in physics, we establish a unified topological theory of PT and CP invariant metals and nodal superconductors, based on the mathematically rigorous KO theory. Representative models are constructed for all nontrivial topological cases in dimensions d=1, 2, and 3, with their exotic physical meanings being elucidated in detail. Intriguingly, it is found that the topological charges of Fermi surfaces in the bulk determine an exotic direction-dependent distribution of topological subgap modes on the boundaries. Furthermore, by constructing an exact bulk-boundary correspondence, we show that the topological Fermi points of the PT and CP invariant classes can appear as gapless modes on the boundary of topological insulators with a certain type of anisotropic crystalline symmetry.

Phonon promoted charge density wave in topological kagome metal ScV6Sn6.

Charge density wave (CDW) orders in vanadium-based kagome metals have recently received tremendous attention, yet their origin remains a topic of debate. The discovery of ScV6Sn6, a bilayer kagome metal featuring an intriguing [Formula: see text] CDW order, offers a novel platform to explore the underlying mechanism behind the unconventional CDW. Here, we combine high-resolution angle-resolved photoemission spectroscopy, Raman scattering and density functional theory to investigate the electronic structure and phonon modes of ScV6Sn6. We identify topologically nontrivial surface states and multiple van Hove singularities (VHSs) in the vicinity of the Fermi level, with one VHS aligning with the in-plane component of the CDW vector near the [Formula: see text] point. Additionally, Raman measurements indicate a strong electron-phonon coupling, as evidenced by a two-phonon mode and new emergent modes. Our findings highlight the fundamental role of lattice degrees of freedom in promoting the CDW in ScV6Sn6.

Edge currents as a signature of flatbands in topological superconductors.

We study nondegenerate flatbands at the surfaces of noncentrosymmetric topological superconductors by exact diagonalization of Bogoliubov-de Gennes Hamiltonians. We show that these states are strongly spin polarized and acquire a chiral dispersion when placed in contact with a ferromagnetic insulator. This chiral mode carries a large edge current which displays a singular dependence on the exchange-field strength. The contribution of other edge states to the current is comparably weak. We hence propose that the observation of the edge current can serve as a test of the presence of nondegenerate flatbands.

Anomalous enhancement of charge density wave in kagome superconductor CsV3Sb5 approaching the 2D limit.

The recently discovered kagome metals AV3Sb5 (A = Cs, Rb, K) exhibit a variety of intriguing phenomena, such as a charge density wave (CDW) with time-reversal symmetry breaking and possible unconventional superconductivity. Here, we report a rare non-monotonic evolution of the CDW temperature (TCDW) with the reduction of flake thickness approaching the atomic limit, and the superconducting transition temperature (Tc) features an inverse variation with TCDW. TCDW initially decreases to a minimum value of 72 K at 27 layers and then increases abruptly, reaching a record-high value of 120 K at 5 layers. Raman scattering measurements reveal a weakened electron-phonon coupling with the reduction of sample thickness, suggesting that a crossover from electron-phonon coupling to dominantly electronic interactions could account for the non-monotonic thickness dependence of TCDW. Our work demonstrates the novel effects of dimension reduction and carrier doping on quantum states in thin flakes and provides crucial insights into the complex mechanism of the CDW order in the family of AV3Sb5 kagome metals.

Tunable topological Dirac surface states and van Hove singularities in kagome metal GdV6Sn6.

Transition-metal-based kagome materials at van Hove filling are a rich frontier for the investigation of novel topological electronic states and correlated phenomena. To date, in the idealized two-dimensional kagome lattice, topologically Dirac surface states (TDSSs) have not been unambiguously observed, and the manipulation of TDSSs and van Hove singularities (VHSs) remains largely unexplored. Here, we reveal TDSSs originating from a ℤ2 bulk topology and identify multiple VHSs near the Fermi level (EF) in magnetic kagome material GdV6Sn6. Using in situ surface potassium deposition, we successfully realize manipulation of the TDSSs and VHSs. The Dirac point of the TDSSs can be tuned from above to below EF, which reverses the chirality of the spin texture at the Fermi surface. These results establish GdV6Sn6 as a fascinating platform for studying the nontrivial topology, magnetism, and correlation effects native to kagome lattices. They also suggest potential application of spintronic devices based on kagome materials.

Berry curvature-induced local spin polarisation in gated graphene/WTe2 heterostructures.

Experimental control of local spin-charge interconversion is of primary interest for spintronics. Van der Waals (vdW) heterostructures combining graphene with a strongly spin-orbit coupled two-dimensional (2D) material enable such functionality by design. Electric spin valve experiments have thus far provided global information on such devices, while leaving the local interplay between symmetry breaking, charge flow across the heterointerface and aspects of topology unexplored. Here, we probe the gate-tunable local spin polarisation in current-driven graphene/WTe2 heterostructures through magneto-optical Kerr microscopy. Even for a nominal in-plane transport, substantial out-of-plane spin accumulation is induced by a corresponding out-of-plane current flow. We present a theoretical model which fully explains the gate- and bias-dependent onset and spatial distribution of the intense Kerr signal as a result of a non-linear anomalous Hall effect in the heterostructure, which is enabled by its reduced point group symmetry. Our findings unravel the potential of 2D heterostructure engineering for harnessing topological phenomena for spintronics, and constitute an important step toward nanoscale, electrical spin control.

Rich nature of Van Hove singularities in Kagome superconductor CsV3Sb5.

The recently discovered layered kagome metals AV3Sb5 (A = K, Rb, Cs) exhibit diverse correlated phenomena, which are intertwined with a topological electronic structure with multiple van Hove singularities (VHSs) in the vicinity of the Fermi level. As the VHSs with their large density of states enhance correlation effects, it is of crucial importance to determine their nature and properties. Here, we combine polarization-dependent angle-resolved photoemission spectroscopy with density functional theory to directly reveal the sublattice properties of 3d-orbital VHSs in CsV3Sb5. Four VHSs are identified around the M point and three of them are close to the Fermi level, with two having sublattice-pure and one sublattice-mixed nature. Remarkably, the VHS just below the Fermi level displays an extremely flat dispersion along MK, establishing the experimental discovery of higher-order VHS. The characteristic intensity modulation of Dirac cones around K further demonstrates the sublattice interference embedded in the kagome Fermiology. The crucial insights into the electronic structure, revealed by our work, provide a solid starting point for the understanding of the intriguing correlation phenomena in the kagome metals AV3Sb5.

4D spinless topological insulator in a periodic electric circuit.

According to the mathematical classification of topological band structures, there exist a number of fascinating topological states in dimensions larger than three with exotic boundary phenomena and interesting topological responses. While these topological states are not accessible in condensed matter systems, recent works have shown that synthetic systems, such as photonic crystals or electric circuits, can realize higher-dimensional band structures. Here, we argue that, because of its symmetry properties, the 4D spinless topological insulator is particularly well suited for implementation in these synthetic systems. We explicitly construct a 2D electric circuit lattice, whose resonance frequency spectrum simulates the 4D spinless topological insulator. We perform detailed numerical calculations of the circuit lattice and show that the resonance frequency spectrum exhibits pairs of 3D Weyl boundary states, a hallmark of the nontrivial topology. These pairs of 3D Weyl states with the same chirality are protected by classical time-reversal symmetry that squares to +1, which is inherent in the proposed circuit lattice. We also discuss how the simulated 4D topological band structure can be observed in experiments.

Phase-resolved Higgs response in superconducting cuprates.

In high-energy physics, the Higgs field couples to gauge bosons and fermions and gives mass to their elementary excitations. Experimentally, such couplings can be inferred from the decay product of the Higgs boson, i.e., the scalar (amplitude) excitation of the Higgs field. In superconductors, Cooper pairs bear a close analogy to the Higgs field. Interaction between the Cooper pairs and other degrees of freedom provides dissipation channels for the amplitude mode, which may reveal important information about the microscopic pairing mechanism. To this end, we investigate the Higgs (amplitude) mode of several cuprate thin films using phase-resolved terahertz third harmonic generation (THG). In addition to the heavily damped Higgs mode itself, we observe a universal jump in the phase of the driven Higgs oscillation as well as a non-vanishing THG above Tc. These findings indicate coupling of the Higgs mode to other collective modes and potentially a nonzero pairing amplitude above Tc.

Scaling theory of [Formula: see text] topological invariants.

For inversion-symmetric topological insulators and superconductors characterized by [Formula: see text] topological invariants, two scaling schemes are proposed to judge topological phase transitions driven by an energy parameter. The scaling schemes renormalize either the phase gradient or the second derivative of the Pfaffian of the time-reversal operator, through which the renormalization group flow of the driving energy parameter can be obtained. The Pfaffian near the time-reversal invariant momentum is revealed to display a universal critical behavior for a great variety of models examined.

Topological surface states in nodal superconductors.

Topological superconductors have become a subject of intense research due to their potential use for technical applications in device fabrication and quantum information. Besides fully gapped superconductors, unconventional superconductors with point or line nodes in their order parameter can also exhibit nontrivial topological characteristics. This article reviews recent progress in the theoretical understanding of nodal topological superconductors, with a focus on Weyl and noncentrosymmetric superconductors and their protected surface states. Using selected examples, we review the bulk topological properties of these systems, study different types of topological surface states, and examine their unusual properties. Furthermore, we survey some candidate materials for topological superconductivity and discuss different experimental signatures of topological surface states.

Dirac surface states and nature of superconductivity in Noncentrosymmetric BiPd.

In non-magnetic bulk materials, inversion symmetry protects the spin degeneracy. If the bulk crystal structure lacks a centre of inversion, however, spin-orbit interactions lift the spin degeneracy, leading to a Rashba metal whose Fermi surfaces exhibit an intricate spin texture. In superconducting Rashba metals a pairing wavefunction constructed from these complex spin structures will generally contain both singlet and triplet character. Here we examine the possible triplet components of the order parameter in noncentrosymmetric BiPd, combining for the first time in a noncentrosymmetric superconductor macroscopic characterization, atomic-scale ultra-low-temperature scanning tunnelling spectroscopy, and relativistic first-principles calculations. While the superconducting state of BiPd appears topologically trivial, consistent with Bardeen-Cooper-Schrieffer theory with an order parameter governed by a single isotropic s-wave gap, we show that the material exhibits Dirac-cone surface states with a helical spin polarization.

Spin-orbital singlet and quantum critical point on the diamond lattice: FeSc2S4.

We present a theory of spin and orbital physics in the A-site spinel compound FeSc2S4, which experimentally exhibits a broad "spin-orbital liquid" regime. A spin-orbital Hamiltonian is derived from a combination of microscopic consideration and symmetry analysis. We demonstrate a keen competition between spin-orbit interactions, which favor formation of a local "spin-orbital singlet," and exchange, which favors magnetic and orbital ordering. Separating the spin-orbital singlet from the ordered state is a quantum critical point. We argue that FeSc2S4 is close to this quantum critical point on the spin-orbital singlet side. The full phase diagram includes a commensurate-incommensurate transition within the ordered phase. A variety of comparisons to and suggestions for experiments are discussed.